We split the proof in two parts.
The \(p<+\infty \)  case.
Let \(0\le s<t\le T\)  be fixed. We denote \(\mathfrak {j} = \min \lbrace j\in \mathbb {N};\;2^{-\frac{j}{H}} \le (t-s)\rbrace  \) . First, remark that
\(\Vert  (T^{\phi ,\varphi } \Delta _j f)_{s,t} \Vert _{L^p(\mathbb {R}^d)}& \le \int _s^t \Vert \phi (r,\cdot )\Delta _j f(r,\varphi (r,\cdot )) \Vert _{L^p(\mathbb {R}^d)} \mathop {}\!\mathrm {d}r \nonumber \\ &\le (t-s) \Vert \phi \Vert _{L^\infty ([0,T]\times \mathbb {R}^d)}\Vert \det (J_{\varphi ^{-1}})\Vert _{L^\infty ([0,T]\times \mathbb {R}^d)} \Vert \Delta _j f\Vert _{L^p(\mathbb {R}^d)}.\) 

For any \(-1\le j < \mathfrak {j}\)  and \(\varepsilon \in [0,1]\) , this yields, since \((t-s)\le 2^{-\frac{j}{H}}\) ,
\(\Vert  (T^{\phi ,\varphi } \Delta _j f)_{s,t} \Vert _{L^p(\mathbb {R}^d)} \le (t-s)^{\frac{1+\varepsilon }{2}} 2^{-\frac{1-\varepsilon }{2H}j} \Vert \phi \Vert _{L^\infty ([0,T]\times \mathbb {R}^d)}\Vert \det (J_{\varphi ^{-1}})\Vert _{L^\infty ([0,T]\times \mathbb {R}^d)} \Vert \Delta _j f\Vert _{L^p(\mathbb {R}^d)}.\) 
Furthermore, thanks to Lemma REF , we also deduce that, for any \(\eta \in [0,1]\) ,
\(\left\Vert \mathbb {E}[(T^{\phi ,\varphi }\Delta _j f)_{s,t}| \mathcal {F}_s ]\right\Vert _{L^p(\mathbb {R}^d)} \hspace{50.0pt}&\nonumber \\\lesssim (t-s)^\eta \Vert \Delta _j f\Vert _{L^p(\mathbb {R}^d)} 2^{-\frac{1-\eta }{H}j} G_{p,s}&\left(1+ \mathbb {E}\left[\Vert \det (J_{\varphi ^{-1}})\Vert _{{L^\infty ([0,T]\times \mathbb {R}^d)}} | \mathcal {F}_s\right]^{\frac{1}{p}}\right).\) 

We now assume that \(j\ge \mathfrak {j}\) . Let \(n_j\ge 1\)  and let us define, for any \(k\in \lbrace 0,\ldots ,n_j\rbrace \) , the following quantities :
\(t_k^j : = \frac{k}{n_j}(t-s) + s\quad \mathrm {and}\quad M^j_k : = \mathbb {E}[(T^{\phi ,\varphi }\Delta _j f)_{s,t} |\mathcal {F}_{t_k^j}].\) 
Hence, \((M^j_k)_{0\le k \le n_j}\)  is a martingale in \(L^p(\mathbb {R}^d)\)  with respect to the filtration \((\mathcal {F}_{t^j_k})_{0\le k\le n_j}\) . Furthermore, it follows from (REF ) that
\(\Vert M^j_0\Vert _{L^p(\mathbb {R}^d)} \lesssim (t-s)^\eta \Vert \Delta _j f\Vert _{L^p(\mathbb {R}^d)} 2^{-\frac{1-\eta }{H}j} G_{p,s}\left(1+ \mathbb {E}\left[\Vert \det (J_{\varphi ^{-1}})\Vert _{L^\infty ([0,T]\times \mathbb {R}^d)} | \mathcal {F}_s\right]^{\frac{1}{p}}\right),\) 
and, in particular, by setting \(\eta = (1+\varepsilon )/2\) , where \(\varepsilon \in [0,1]\) , and for \(q\ge 1\) , we have
\(\mathbb {E}[\Vert M^j_0\Vert _{L^p(\mathbb {R}^d)}^q] \lesssim (t-s)^{\frac{1+\varepsilon }{2}} \Vert \Delta _j f\Vert _{L^p(\mathbb {R}^d)}^q 2^{-\frac{1-\varepsilon }{2H}j}.\) 
Since we are working in \(L^p(\mathbb {R}^d)\) , \(p\in [2,+\infty )\) , which is a UMD space of type 2 (see for example [1]}, [2]} and Appendix REF ), we know that the Burkolder-Davis-Gundy inequality holds, and we have, for all \(q\ge 1\) ,
\(\mathbb {E}[\Vert M^j_{n_j} - M^j_{0}\Vert _{L^p(\mathbb {R}^d)}^q] \lesssim \mathbb {E}\left[\left(\sum _{k=0}^{n_j-1}\Vert M^j_{k+1}-M^j_k\Vert ^2_{L^p(\mathbb {R}^d)} \right)^{\frac{q}{2}}\right].\) 
Furthermore, we obtain that, for all \(0\le k\le n_{j}-1\) ,
\(M^j_{k+1}-M^j_k = (T^{\phi ,\varphi }\Delta _j f)_{t^j_k,t^j_{k+1}} + \mathbb {E}\left[(T^{\phi ,\varphi }\Delta _j f)_{t^j_{k+1},t}\Big | \mathcal {F}_{t^j_{k+1}}\right] - \mathbb {E}\left[(T^{\phi ,\varphi }\Delta _j f)_{t^j_{k},t}\Big | \mathcal {F}_{t^j_{k}}\right].\) 
Using again (REF ) to estimate the first term on the right-hand-side and (REF ) for the second and third terms, we deduce, for any \(\varepsilon \in [0,1]\) ,
\(\Vert M^{j}_{k+1}-M^{j}_{k}\Vert _{L^p(\mathbb {R}^d)}\lesssim \Vert \Delta _j f\Vert _{L^p(\mathbb {R}^d)} (t-s)^\varepsilon \left( \frac{(t-s)^{1-\varepsilon }}{n_j} + 2^{-\frac{1-\varepsilon }{H}j}\right) \kappa ^j_{k},\) 
where
\(\kappa ^j_{k} &: =\Vert \phi \Vert _{L^\infty ([0,T]\times \mathbb {R}^d)}\Vert \det (J_{\varphi ^{-1}})\Vert _{L^\infty ([0,T]\times \mathbb {R}^d)}\\ &\hspace{10.0pt}+ G_{p,t^j_k}\left(1+ \mathbb {E}\left[\Vert \det (J_{\varphi ^{-1}})\Vert _{L^\infty ([0,T]\times \mathbb {R}^d)} | \mathcal {F}_{t^j_k}\right]^{\frac{1}{p}}\right)\\ &\hspace{10.0pt}+ G_{p,t^j_{k+1}}\left(1+ \mathbb {E}\left[\Vert \det (J_{\varphi ^{-1}})\Vert _{L^\infty ([0,T]\times \mathbb {R}^d)} | \mathcal {F}_{t^j_{k+1}}\right]^{\frac{1}{p}}\right).\) 

Note that thanks to the hypothesis,
\(\sup _{j\ge 0} \sup _{k\in \lbrace 0,\ldots ,n_j\rbrace } \mathbb {E}[(\kappa ^j_k)^q] <+\infty .\) 
Hence, we have, for any \(q\ge 2\) ,
\(\mathbb {E}[\Vert M^j_{n_j} - M^j_{0}\Vert _{L^p(\mathbb {R}^d)}^q]&\lesssim \mathbb {E}\left[\left(\sum _{k=0}^{n_j-1}\Vert M^j_{k+1}-M^j_k\Vert ^2_{L^p(\mathbb {R}^d)} \right)^{\frac{q}{2}}\right] \\&\lesssim (t-s)^{\varepsilon q} \Vert \Delta _j f\Vert ^q_{L^{p}(\mathbb {R}^d) }\left( \frac{(t-s)^{1-\varepsilon }}{n_j} + 2^{-\frac{1-\varepsilon }{H}j}\right)^q \mathbb {E}\left[\left( \sum _{k=0}^{n_j-1} (\kappa _k^j)^2 \right)^{\frac{q}{2}} \right]\\ &\lesssim (t-s)^{\varepsilon q} \Vert \Delta _j f\Vert ^q_{L^{p}(\mathbb {R}^d)}\left( \frac{(t-s)^{1-\varepsilon }}{n_j} + 2^{-\frac{1-\varepsilon }{H}j}\right)^q \left(\sum _{k=0}^{n_j-1} \mathbb {E}[(\kappa ^j_k)^q]^{\frac{2}{q}}\right)^{\frac{q}{2}}\\ &\lesssim (t-s)^{\varepsilon q} \Vert \Delta _j f\Vert ^q_{L^{p}(\mathbb {R}^d)}\left( \frac{(t-s)^{1-\varepsilon }}{\sqrt{n_j}} + \sqrt{n_j}2^{-\frac{1-\varepsilon }{H}j}\right)^q.\) 

We now optimize the estimate on \(n_j\ge 1\) . We choose
\( n_j = \frac{(t-s)^{1-\varepsilon }}{2^{-\frac{1-\varepsilon }{H}j}} + \iota ,\) 
for a certain \(\iota \in [0,1]\) . This yields, since \(2^{-\frac{j}{H}}\le (t-s)\) ,
\(\sqrt{n_j}2^{-\frac{1-\varepsilon }{H}j} = (t-s)^{\frac{1-\varepsilon }{2}}2^{-\frac{1-\varepsilon }{2H}j}\sqrt{1 + \iota \frac{2^{-\frac{1-\varepsilon }{H}j}}{(t-s)^{1-\varepsilon }}} \le \sqrt{2} (t-s)^{\frac{1-\varepsilon }{2}}2^{-\frac{1-\varepsilon }{2H}j},\) 
and, also,
\(\frac{(t-s)^{1-\varepsilon }}{\sqrt{n_j}} \le (t-s)^{\frac{1-\varepsilon }{2}}2^{-\frac{1-\varepsilon }{2H}j}.\) 
Thus, we deduce that
\(\mathbb {E}\left[\Vert M^j_{n_j} - M^j_{0}\Vert _{L^p(\mathbb {R}^d)}^q\right]^{\frac{1}{q}} \lesssim (t-s)^{\frac{1+\varepsilon }{2}} \Vert  \Delta _j f \Vert _{L^p(\mathbb {R}^d)} 2^{-\frac{(1-\varepsilon )}{2H} j}.\) 
Using the fact that \((T^{\phi ,\varphi }\Delta _j f)_{s,t} = M^j_{n_j}\)  as well as the estimate (REF ), we obtain, for any \(q\ge 2\)  and for all \(j\ge \mathfrak {j}\) ,
\(\mathbb {E}\left[\left\Vert (T^{\phi ,\varphi }\Delta _j f)_{s,t}\right\Vert _{L^p(\mathbb {R}^d)}^q\right]^{\frac{1}{q}} \lesssim (t-s)^{\frac{1+\varepsilon }{2}} \Vert \Delta _j f\Vert _{L^p(\mathbb {R}^d)} 2^{-j\frac{1-\varepsilon }{2H}}.\) 
It follows from (REF ) that the previous estimate holds for all \(\varepsilon \in [0,1]\)  and any \(j\ge -1\) . Thus, we deduce that, for any \(p\in [2,+\infty )\) , \(q\ge 2\)  and \(\varepsilon \in [0,1]\) ,
\(\mathbb {E}\left[\left\Vert (T^{\phi ,\varphi }f)_{s,t}\right\Vert _{L^p(\mathbb {R}^d)}^q\right]^{\frac{1}{q}} &\le \sum _{j\ge -1} \mathbb {E}\left[\left\Vert (T^{\phi ,\varphi }\Delta _j f)_{s,t}\right\Vert _{L^p(\mathbb {R}^d)}^q\right]^{\frac{1}{q}}\\ &\lesssim (t-s)^{\frac{1+\varepsilon }{2}} \sum _{j=-1}^{+\infty } 2^{-j\frac{1-\varepsilon }{2H}} \Vert \Delta _j f\Vert _{L^p(\mathbb {R}^d)} =(t-s)^{\frac{1+\varepsilon }{2}} \Vert f\Vert _{B^{-\frac{1-\varepsilon }{2H}}_{p,1}}.\) 

It follows from Theorem REF  that, for all \(0<\varepsilon ^{\prime }<\varepsilon \)  and \(f\in B^{-\frac{1-\varepsilon }{2H}}_{p,1}\) , we have
\(\Vert T^{\phi ,\varphi }f\Vert _{L^q(\Omega ;C^{\nu }([0,T];L^p(\mathbb {R}^d)))} &\lesssim \mathbb {E}\left[ \int _{[0,T]^2} \frac{\Vert (T^{\phi ,\varphi }f)_{s,t}\Vert _{L^p(\mathbb {R}^d)}^q}{|t-s|^{q\frac{1+\varepsilon ^{\prime }}{2}+1}} \mathop {}\!\mathrm {d}s \mathop {}\!\mathrm {d}t \right]^{1/q}\\ &\lesssim \Vert f\Vert _{B^{-\frac{1-\varepsilon }{2H}}_{p,1}} \int _{[0,T]^2}|t-s|^{q\frac{\varepsilon -\varepsilon ^{\prime }}{2}-1}\mathop {}\!\mathrm {d}s \mathop {}\!\mathrm {d}t \lesssim \Vert f\Vert _{B^{-\frac{1-\varepsilon }{2H}}_{p,1}},\) 

with \(\nu = \frac{1+\varepsilon ^{\prime }}{2}-\frac{1}{q}\) , which is exactly the theorem in the case \(p<+\infty \)  when we remind that whenever \(f\in B^{\alpha }_{p,p}\) , one has for \(\varepsilon ^{\prime \prime }>0\) , thanks to Hölder inequality,
\(\Vert f\Vert _{B^{\alpha -\varepsilon ^{\prime \prime }}_{p,1}} = \sum _{j\ge 1} 2^{j(\alpha -\varepsilon ^{\prime \prime })j} \Vert \Delta _j f\Vert _{L^p(\mathbb {R}^d)} \le \left(\sum _{j\ge -1}2^{-\varepsilon ^{\prime \prime } \frac{r}{r-1}j}\right)^{1-\frac{1}{r}}\left(\sum _{j\ge -1 } 2^{\alpha r j} \Vert \Delta _j f\Vert ^r_{L^p(\mathbb {R}^d)}\right)^{\frac{1}{r}} \\ \lesssim _{\varepsilon ^{\prime \prime }} \Vert f\Vert _{B^\alpha _{p,r}}.\) 

The \(p=+\infty \)  case.
The proof is pretty much the same as the proof in the previous case. We provide the main arguments for completeness. Let \(0\le s<t\le T\)  be fixed. We denote \(\mathfrak {j} = \min \lbrace j\in \mathbb {N};\;2^{-\frac{j}{H}} \le (t-s)\rbrace  \) . First, we can see that, for any \(-1\le j < \mathfrak {j}\)  and \(\varepsilon \in [0,1]\) , this yields, since \((t-s)\le 2^{-\frac{j}{H}}\) , for any \(x\in \mathbb {R}^d\) ,
\(\left| (T^{\phi ,\varphi } \Delta _j f)_{s,t}(x) \right| \le (t-s)^{\frac{1+\varepsilon }{2}} 2^{-\frac{1-\varepsilon }{2H}j} \Vert \phi \Vert _{L^\infty ([0,T]\times \mathbb {R}^d)} \Vert \Delta _j f\Vert _{L^{\infty }(\mathbb {R}^d)}.\) 
Moreover, it follows from Lemma REF  that, for any \(\eta \in [0,1]\) ,
\(\left|\mathbb {E}[(T^{\phi ,\varphi }\Delta _j f)_{s,t}(x)| \mathcal {F}_s ]\right|\lesssim (t-s)^\eta \Vert \Delta _j f\Vert _{L^{\infty }(\mathbb {R}^d)} 2^{-\frac{1-\eta }{H}j} G_{\infty ,s}.\) 

We now assume that \(j\ge \mathfrak {j}\) . Let \(x\in \mathbb {R}^d\) , \(n_j\ge 1\)  and let us define, for any \(k\in \lbrace 0,\ldots ,n_j\rbrace \) , the following quantities :
\(t_k^j : = \frac{k}{n_j}(t-s) + s\quad \mathrm {and}\quad M^j_k(x) : = \mathbb {E}[(T^{\phi ,\varphi }\Delta _j f)_{s,t}(x) |\mathcal {F}_{t_k^j}].\) 
We can see that, \((M^j_k(x))_{0\le k \le n_j}\)  is a martingale with respect to the filtration \((\mathcal {F}_{t^j_k})_{0\le k\le n_j}\) . Thanks to (REF ), we obtain
\(|M^j_0(x)| \lesssim (t-s)^\eta \Vert \Delta _j f\Vert _{L^{\infty }(\mathbb {R}^d)} 2^{-\frac{1-\eta }{H}j} G_{\infty ,s},\) 
which yields, by setting \(\eta = (1+\varepsilon )/2\) , where \(\varepsilon \in [0,1]\) , and for \(q\ge 1\) ,
\(\mathbb {E}[|M^j_0(x)|^q] \lesssim (t-s)^{\frac{1+\varepsilon }{2}} \Vert \Delta _j f\Vert _{L^{\infty }(\mathbb {R}^d)}^q 2^{-\frac{1-\varepsilon }{2H}j}.\) 
It follows from (REF ) and (REF ) that, for any \(\varepsilon \in [0,1]\)  and \(1\le k\le n_j\) ,
\(|M^{j}_{k+1}(x)-M^{j}_{k}(x)|\lesssim \Vert \Delta _j f\Vert _{L^{\infty }(\mathbb {R}^d)} (t-s)^\varepsilon \left( \frac{(t-s)^{1-\varepsilon }}{n_j} + 2^{-\frac{1-\varepsilon }{H}j}\right) \kappa ^j_{k},\) 
where
\(\kappa ^j_{k} : =\Vert \phi \Vert _{L^\infty ([0,T]\times \mathbb {R}^d)}+ G_{\infty ,t^j_k}+ G_{\infty ,t^j_{k+1}}.\) 
The assumptions gives the bound
\(\sup _{j\ge 0} \sup _{k\in \lbrace 0,\ldots ,n_j\rbrace } \mathbb {E}[(\kappa ^j_k)^q] <+\infty .\) 
From here, by following the same arguments as in the \(p<+\infty \)  case, we deduce, thanks to the BDG inequality and some optimization on \(n_j\) , that, for any \(q\ge 2\) , for any \(\varepsilon \in [0,1]\)  and for all \(j\ge -1\) ,
\(\mathbb {E}\left[|(T^{\phi ,\varphi }\Delta _j f)_{s,t}(x)|^q\right]^{\frac{1}{q}} \lesssim (t-s)^{\frac{1+\varepsilon }{2}} \Vert \Delta _j f\Vert _{L^{\infty }(\mathbb {R}^d)} 2^{-j\frac{1-\varepsilon }{2H}},\) 
and we obtain the estimate, for any \(x\in \mathbb {R}^d\) ,
\(\mathbb {E}[|(T^{\phi ,\varphi } f)_{s,t}(x)|^q]^{\frac{1}{q}} \lesssim |t-s|^{\frac{1+\varepsilon }{2}} \Vert f\Vert _{B^{-\frac{1-\varepsilon }{2H}}_{\infty ,1}}.\) 
Finally, Theorem REF  gives the desired result.
We can extend the previous result with the help of the following Lemma.
Lemme 3.5 
Let \(\beta \in \mathbb {N}^d\) , \(\phi ,\varphi \in L^q(\Omega ; L^{\infty }([0,T];C^{|\beta |}(\mathbb {R}^d)))\)  and \(f\in \mathcal {S}\) . Then, we have, for any \((t,x)\in [0,T]\times \mathbb {R}^d\) ,
\(\partial ^\beta (T^{\phi ,\varphi })_t f(x)=\sum _{\gamma \le \beta } \sum _{\begin{array}{c}1\le |\mu |\le |\gamma |\\ \upsilon \in \mathcal {N}_{\gamma ,\mu }\end{array}} C_{\beta ,\gamma ,\mu ,\upsilon } \left(T^{\Lambda _{\gamma ,\upsilon }(\varphi )\partial ^{\beta -\gamma }\phi , \varphi } \big (\partial ^{\mu } f\big )\right)_t(x),\) 
where \(\Lambda _{\gamma ,\upsilon }(\varphi ) : = \prod _{\begin{array}{c}1\le |\delta |\le |\gamma |\\1\le j \le d\end{array}} (\partial ^{\delta } \varphi _{j}(r,x))^{\upsilon _{\delta _j}}\)  and \(C_{\beta ,\gamma ,\mu ,\upsilon }>0\)  and
\(\mathcal {N}_{\gamma ,\mu } : = \left\lbrace  \upsilon \in \mathbb {N}^d;\;\sum _{1\le |\delta |\le |\gamma |} \upsilon _{\delta _j} = \mu _j,\;\textrm {for any}\; 1\le j\le d,\;\mathrm {and}\;\sum _{\begin{array}{c}1\le |\delta |\le |\gamma |\\1\le j \le d\end{array}} \delta \upsilon _{\delta _j} = \gamma \right\rbrace ,\) 

The proof of the lemma is a straightforward consequence of Leibniz's rule of differentiation, which yields
\(\partial ^{\beta } \phi (r,x) b(\varphi (r,x)) = \sum _{\gamma \le \beta } C^1_{\beta ,\gamma } \partial ^{\beta -\gamma }\phi (r,x) \partial ^{\gamma } (b(\varphi (r,x))),\) 
for some constants \(\lbrace C^1_{\beta ,\gamma }\rbrace _{\gamma \le \beta }\subset \mathbb {R}^+\)  and Faà di Bruno's formula, which gives
\(\partial ^{\gamma } (b(\varphi (r,x))) = \sum _{\begin{array}{c}1\le |\mu |\le |\gamma |\\ \upsilon \in \mathcal {N}_{\gamma ,\mu }\end{array}} C^2_{\mu ,\upsilon } \partial ^{\mu } b (\varphi (r,x)) \prod _{\begin{array}{c}1\le |\delta |\le |\gamma |\\1\le j \le d\end{array}} (\partial ^{\delta } \varphi _{j}(r,x))^{\upsilon _{\delta _j}},\) 
for some constants \(\lbrace C^2_{\upsilon ,\mu }\rbrace _{\begin{array}{c}1\le |\mu |\le |\gamma |\\ \upsilon \in \mathcal {N}_{\gamma ,\mu }\end{array}}\subset \mathbb {R}^+\) .
We can now state a by-product of Theorem REF .
Corollary 3.6 
Let \(p\in [2,+\infty ]\) , \(r\in [1,+\infty ]\) , \(n\ge 1\)  and \((\phi ,\varphi )\)  be a couple of functions such that, for any \(\beta ,\gamma ,\mu ,\upsilon \in \mathbb {N}^d\)  that verify \(|\beta |\le n\) , \(\gamma \le \beta \) , \(1\le |\mu |\le |\gamma |\)  and \(\upsilon \in \mathcal {N}_{\gamma ,\mu }\) , \((\Lambda _{\gamma ,\upsilon }(\varphi )\partial ^{\beta -\gamma }\phi ,\varphi )\)  satisfies Assumption REF .
Then, for all \(\varepsilon \in (0,1]\) , \(\varepsilon ^{\prime },\varepsilon ^{\prime \prime }\in (0,\varepsilon )\) , \(\kappa \in (0,n]\) , \(f\in B^{\kappa -\frac{1-\varepsilon }{2H}}_{p,r}\) , \(\eta > d/q\)  and \(q\ge 2\)  such that \(q \frac{1+\varepsilon ^{\prime }}{2}>1\) , \(\nu = \frac{1+\varepsilon ^{\prime }}{2}-\frac{1}{q}\) , the following estimate holds
\(\Vert T^{\phi ,\varphi } f\Vert _{L^q(\Omega ; \mathcal {C}^{\nu }([0,T];E_{p,\kappa }))} \lesssim \Vert f\Vert _{B^{\kappa -\frac{1-\varepsilon ^{\prime \prime }}{2H}}_{p,r}},\) 
with \(E_{p,\kappa } = B^{\kappa }_{p,p}(\mathbb {R}^d)\)  if \(p<+\infty \)  and \(E_{\infty ,\kappa }=\mathcal {C}^{\kappa }_w(\mathbb {R}^d)\)  if \(p = +\infty \) , where \(w\)  is a weight with \(\eta \) -polynomial growth. When \(p<+\infty \)  and \(r=1\) , one can take \(\varepsilon ^{\prime \prime }=\varepsilon \) .

As usual, one needs to distinguish between \(p=+\infty \)  and \(1<p<+\infty \) .
The \(p<+\infty \)  case.
Thanks to Lemma REF  and (REF ), we obtain that, for any \(\beta \in \mathbb {N}^d\) , \(\varepsilon \in [0,1]\) , \(q\ge 2\)  and \(j\ge -1\) ,
\(\mathbb {E}\left[\left\Vert \partial ^{\beta }(T^{\phi ,\varphi }\Delta _j f)_{s,t}\right\Vert _{L^p(\mathbb {R}^d)}^q\right]^{\frac{1}{q}}&\le \sum _{\gamma \le \beta } \sum _{\begin{array}{c}1\le |\mu |\le |\gamma |\\ \upsilon \in \mathcal {N}_{\gamma ,\mu }\end{array}} C_{\beta ,\gamma ,\mu ,\upsilon }\mathbb {E}\left[\left\Vert T^{\Lambda _{\gamma ,\upsilon }(\varphi )\partial ^{\beta -\gamma }\phi , \varphi } \big (\partial ^{\mu } \Delta _j f\big ))\right\Vert _{L^p(\mathbb {R}^d)}^q\right]^{\frac{1}{q}}\\ &\lesssim (t-s)^{\frac{1+\varepsilon }{2}} 2^{-j\frac{1-\varepsilon }{2H}} \sum _{\gamma \le \beta } \Vert \partial ^{\gamma }\Delta _j f\Vert _{L^p(\mathbb {R}^d)}\\ &\lesssim (t-s)^{\frac{1+\varepsilon }{2}} 2^{-j\frac{1-\varepsilon }{2H}+|\beta |j} \Vert \Delta _j f\Vert _{L^p(\mathbb {R}^d)}.\) 

In particular, we deduce that, for any \(k\in \mathbb {N}\) , we have
\(\mathbb {E}\left[\left\Vert (T^{\phi ,\varphi }\Delta _j f)_{s,t}\right\Vert _{B_{p,p}^{2k}}^q\right]^{\frac{1}{q}} &\simeq \mathbb {E}\left[\left\Vert (1-\Delta )^{2k}(T^{\phi ,\varphi }\Delta _j f)_{s,t}\right\Vert _{L^{p}(\mathbb {R}^d)}^q\right]^{\frac{1}{q}}\\ &\lesssim (t-s)^{\frac{1+\varepsilon }{2}} 2^{-j\frac{1-\varepsilon }{2H}+2kj} \Vert \Delta _j f\Vert _{L^p(\mathbb {R}^d)},\) 

and, thus, by interpolation, we deduce that, for any \(\varrho \in \mathbb {R}\) ,
\(\mathbb {E}\left[\left\Vert (T^{\phi ,\varphi }\Delta _j f)_{s,t}\right\Vert _{B_{p,p}^{\varrho }}^q\right]^{\frac{1}{q}}\lesssim (t-s)^{\frac{1+\varepsilon }{2}} 2^{-j\frac{1-\varepsilon }{2H}+\varrho j} \Vert \Delta _j f\Vert _{L^p(\mathbb {R}^d)}.\) 
In particular, this yields
\(\mathbb {E}\left[\left\Vert (T^{\phi ,\varphi }f)_{s,t}\right\Vert _{W^{2k,p}(\mathbb {R}^d)}^q\right]^{\frac{1}{q}}\lesssim (t-s)^{\frac{1+\varepsilon }{2}} \Vert f\Vert _{B^{-\frac{1-\varepsilon }{2H}+2k}_{p,1}},\) 
which gives the desired result by using Kolmogorov's continuity theorem.

The \(p=+\infty \)  case.
For any \(\beta \in \mathbb {N}^d\) , any \(x,y\in \mathbb {R}^d\) , we have, for any \(\theta \in [0,1]\)  and \(j\ge -1\) ,
\(&\left|\partial ^{\beta }(T^{\phi ,\varphi }\Delta _j f)_{s,t}(x) - \partial ^{\beta }(T^{\phi ,\varphi }\Delta _jf)_{s,t}(y)\right|\\ &\hspace{30.0pt} \le |\nabla \partial ^{\beta }(T^{\phi ,\varphi }\Delta _j f)_{s,t}(z)|^{\theta } \left(\left|\partial ^{\beta }(T^{\phi ,\varphi }\Delta _j f)_{s,t}(x)\right| + \left|\partial ^{\beta }(T^{\phi ,\varphi }\Delta _j f)_{s,t}(y)\right|\right)^{1-\theta } |x-y|^{\theta },\) 

for some \(z\in \lbrace (1-\zeta )x + \zeta y:\zeta \in [0,1]\rbrace \) . It follows from Lemma REF  and (REF ) that, for any \(q\ge 2\) ,
\(\mathbb {E}\left[\left|\partial ^{\beta }(T^{\phi ,\varphi }\Delta _j f)_{s,t}(x) - \partial ^{\beta }(T^{\phi ,\varphi } \Delta _jf)_{s,t}(y)\right|^q\right]^{1/q} \le |x-y|^{\theta } \mathbb {E}\left[\left|\nabla \partial ^{\beta }(T^{\phi ,\varphi }\Delta _j f)_{s,t}(z)\right|^{2\theta q}\right]^{1/(2q)}\\ \times \left(\mathbb {E}\left[\left| \partial ^{\beta }(T^{\phi ,\varphi }\Delta _j f)_{s,t}(x)\right|^{2(1-\theta )q}\right]^{1/(2(1-\theta )q)}+\mathbb {E}\left[\left| \partial ^{\beta }(T^{\phi ,\varphi }\Delta _j f)_{s,t}(y)\right|^{2(1-\theta )q}\right]^{1/(2(1-\theta )q)} \right)^{1-\theta }\\ \lesssim |x-y|^{\theta } (t-s)^{\frac{1+\varepsilon }{2}}2^{-j\frac{1-\varepsilon }{2H}}\left(\sum _{\gamma \le \beta } \Vert \nabla \partial ^{\gamma }\Delta _j f\Vert _{L^{\infty }(\mathbb {R}^d)}\right)^{1-\theta } \left(\sum _{\gamma \le \beta } \Vert \partial ^{\gamma }\Delta _j f\Vert _{L^{\infty }(\mathbb {R}^d)}\right)^{1-\theta }.\\ \lesssim |x-y|^{\theta } (t-s)^{\frac{1+\varepsilon }{2}}2^{-j\frac{1-\varepsilon }{2H}}2^{(|\beta | + \theta )j}\Vert \Delta _j f\Vert _{L^{\infty }(\mathbb {R}^d)}.\) 

We also have, by Lemma REF  and (REF ),
\(\mathbb {E}\left[\left|\partial ^{\beta }(T^{\phi ,\varphi }\Delta _j f)_{s,t}(x)\right|^q\right]^{1/q} \lesssim (t-s)^{\frac{1+\varepsilon }{2}}2^{-j\frac{1-\varepsilon }{2H}}2^{(|\beta | + \theta )j}\Vert \Delta _j f\Vert _{L^{\infty }(\mathbb {R}^d)}.\) 
From here, we use Theorem REF  and Theorem REF  to deduce the desired result.

Malliavin Calculus
In order to be self-contain, and in the spirit of Section  we recall in this section some facts about Malliavin calculus. Most of them are well-known facts, but Subsection
refsubsec:conditional yields to non-standard estimates that we will use in the following. In particular, we will derived a not so surprising conditional integration by part formula, which will allow us to check whenever a flow generated by a RDE driven by a Gaussian rough path satisfy the conditions of Theorem REF  or Corollary REF .
One may consult
[3]}, [4]} for more details.
Isonormal Gaussian processes
Definition 4.1 (Isonormal Gaussian process) An Isonormal Gaussian process is the set of:

a real and separable Hilbert space \(\mathcal {H}\)  whose scalar product is denoted as \(\langle \cdot ,\cdot \rangle _{\mathcal {H}}\)  and its norm as \(\Vert \cdot \Vert _\mathcal {H}\) ,

a complete probability space \((\Omega ,\mathcal {F},\mathbb {P})\) ,

a real-valued Gaussian process \(W:h\in \mathcal {H}\rightarrow W(h)\) , i.e. \((W(h))_{h\in \mathcal {H}}\)  is a family of centered Gaussian random variables such that \(\mathbb {E}[W(h)W(g)] = \langle h,g\rangle _\mathcal {H}\) , for any \(h,g\in \mathcal {H}\) .


Remark 4.2

By Kolmogorov's theorem, given only \(\mathcal {H}\) , we can construct \((\Omega ,\mathcal {F},\mathbb {P})\)  and \(W\)  satisfying the above conditions.

The mapping \(h\rightarrow W(h)\)  is linear.


The two following examples will be of interest in the following. The first one is a classical construction linked to the standard Brownian motion.
Example 4.3 (Isonormal Gaussian process associated to the Brownian motion) Let \((B_t)_{t\ge 0}\)  be a \(d\) -dimensional Brownian motion defined on its canonical probability space \((\Omega ,\mathcal {F},\mathbb {P})\) . Let \(\mathcal {H}= L^2(\mathbb {R}^+;\mathbb {R}^d)\)  and, for any \(h\in \mathcal {H}\) , let us define \(W(h)\)  as the Wiener integral
\(W(h) = \sum _{k = 1}^d \int _0^{+\infty } h_k(s) \mathop {}\!\mathrm {d}B^{k}_s.\) 

The second example deals with a generic construction of an isonormal Gaussian process associated to a generic Gaussian process. This example will be of primer importance in the following, as we will always work in this specific setting.
Example 4.4 Let us consider any vector-valued continuous centered Gaussian process \(W = (W^1,\cdots ,W^n)\)  with iid components and let us construct a corresponding (vector-valued) isonormal process. Fix \(j\in \lbrace 1,\ldots ,d\rbrace \) , \(d\ge 1\) . We denote the covariance function of \(W\)  as
\(R_W(t,s) : = \mathbb {E}\left[W_t^{j}W_s^{)} \right].\) 
We now consider \(\mathcal {E}\)  the set of step functions on \([0,1]\)
\(\mathcal {E}: = \left\lbrace f = \sum _{k = 1}^{m} a_k\mathbb {1}_{[t_{k},t_{k+1}]};\; m\in \mathbb {N}^*,\, (a_k)_{1\le k\le m}\in \mathbb {R}^{m-1},\, (t_k)_{1\le k\le m+1}\in \Pi ([0,1]) \right\rbrace .\) 
We define \(\mathcal {H}\)  as the closure of \(\mathcal {E}\)  with respect to the scalar product
\(\langle \mathbb {1}_{[0,t]},\mathbb {1}_{[0,s]}\rangle _\mathcal {H}= R_W(t,s),\) 
which leads to the norm defined on \(\mathcal {E}\)  by
\(\Vert f\Vert _{\mathcal {H}}^2 = \sum _{k,\ell = 1}^m a_k a_{\ell } \langle \mathbb {1}_{[0,t_k]}, \mathbb {1}_{[0,t_{\ell }]}\rangle _{\mathcal {H}} = \sum _{k,\ell = 1}^m a_k a_{\ell }R_W(t_{k},t_{\ell }).\) 
We can extend the linear mapping \(\mathcal {K}:\mathbb {1}_{[0,t]}\in \mathcal {E}\rightarrow W^{j}(t)\in L^2(\Omega )\)  to an isometry \(\mathcal {K}:h\in \mathcal {H}\rightarrow W^{j}(h)\)  since
\(\mathbb {E}\left[ \mathcal {K}(\mathbb {1}_{[0,t]}) \mathcal {K}(\mathbb {1}_{[0,s]})\right]= \mathbb {E}\left[W^{j}_t \cdot W^{j}_s\right] = R_W(t,s) = \langle \mathbb {1}_{[0,t]}, \mathbb {1}_{[0,s]}\rangle _{\mathcal {H}}.\) 
Then, \(\mathcal {K}:h\in \mathcal {H}\rightarrow W^{j}(h)\)  is an isonormal Gaussian process. Let us construct the vector-valued isonormal Gaussian process \(\mathcal {K}:h\in \mathcal {H}^{\oplus d} \rightarrow W(h) = (W^{1}(h_1),\cdots ,W^{n}(h_n))\in L^2(\Omega ;\mathbb {R}^d)\) . In the following, we will treat the case \(n = 1\)  without loss of generality.

Remark 4.5

An interesting instance is, for example, the case where \(W = B^H\)  the fractional Brownian motion with Hurst parameter \(H\in (0,1)\)  and covariance
\(R_{B^H}(t,s) = \mathbb {E}[B^H_tB^H_s] = \frac{1}{2}\left( t^{2H}+s^{2H} - |t-s|^{2H}\right).\) 

For \(W_0 =0\)  (and \(R_W(0,0) = 0\) ), we have, by [5]},
\(\langle h_1,h_2\rangle _{\mathcal {H}} = \int _0^1\int _0^1 h_1(t)h_2(s)\mathop {}\!\mathrm {d}R_W(t,s),\) 
whenever the 2d Young integral on the right-hand-side is well-defined.


Since \(\mathcal {H}\)  is the closure of indicator functions, for any \([a,b]\subset [0,1]\) , we can also define \(\mathcal {H}([a,b])\)  the restriction of \(\mathcal {H}= \mathcal {H}([0,1])\)  to \([a,b]\) . Thus, for almost every \(s\in [a,b]^c\) , \(h(s) = 0\)  for \(h\in \mathcal {H}([a,b])\) . Furthermore, we directly deduce that, for any \(h,g\in \mathcal {H}\) ,
\(\langle h, g\rangle _{\mathcal {H}([a,b])} = \langle h\mathbb {1}_{[a,b]}, g\mathbb {1}_{[a,b]}\rangle _{\mathcal {H}}.\) 
For any \([a,b]\subset [0,1]\) , we denote \(\mathcal {F}_{[a,b]}\)  the \(\sigma \) -algebra generated by \(\lbrace W(\mathbb {1}_{[u,v]}),\, [u,v]\in [a,b]\rbrace \) .

The Malliavin derivative
We consider the set of smooth cylindrical fields \(S\)  given by
\(S = \lbrace F = F(W) = f(W(h_1),W(h_2),\ldots ,W(h_n)),\;f\in C^{\infty }_p(\mathbb {R}^d),\; (h_k)_{k\in \lbrace 1,\ldots ,n\rbrace }\in \mathcal {H}^n,\; n\ge 0\rbrace ,\) 
where \(\mathcal {C}^{\infty }_p(\mathbb {R}^d)\)  denotes the set of functions from \(\mathbb {R}^d\)  to \(\mathbb {R}\)  that are infinitely differentiable and with all their partial derivatives having polynomial growth. Furthermore, we denote \(C^{\infty }_b(\mathbb {R}^d)\)  the subset of \(C^{\infty }_p(\mathbb {R}^d)\)  where all the partial derivatives are bounded. The smooth random variables associated to this space \(C^\infty _b\)  is denoted by \( S_b\) . We note that \( S_b\)  (and \(S\) ) is dense in \(L^2(\Omega )\) .
Definition 4.6 For any \(F=f\big (W(h_1)\cdots W(h_n)\big )\in S\)  with \(f\in C^\infty _p\) , we define the Malliavin derivative of \(F\)  as
\(DF = \sum _{k = 1}^n \partial _k f(W(h_1),\ldots ,W(h_n)) h_k,\) 
which takes values in \(\mathcal {H}\) .

We can directly check that, for any \(F,G\in S\) , we have the "chain rule" relation
\(D(FG) = G DF + F DG.\) 
One can reformulate this definition in terms of of directional derivative in the so called Cameron-Martin space. Indeed, for isonormal Gaussian process \(W\) , any \(h\in \mathcal {H}\)  and \(\varepsilon \in [0,1]\) , we defined the shifted process
\(\tau _{\varepsilon ,h}W : u\in \mathcal {H}\rightarrow W(u) + \varepsilon \langle u,h\rangle _{\mathcal {H}}.\) 
Then, the Malliavin derivative can be identified as
\(\langle DF,h\rangle _{\mathcal {H}} = \lim _{\varepsilon \rightarrow 0} \varepsilon ^{-1}\left(F(\tau _{\varepsilon ,h}W) - F(W)\right),\quad \forall h\in \mathcal {H},\) 
that is an element in the dual space \(\mathcal {H}^{\prime }\)  (which is still \(\mathcal {H}\)  here).
The shifted process \(\tau _{\varepsilon ,h}W\)  can also be interpreted for \(h\)  in the so-called Cameron-Martin space denoted \(\mathcal {H}_1\)  which is defined as the completion of
\(\tilde{\mathcal {E}} = \left\lbrace  t\rightarrow \sum _{k = 1}^m a_kR_W(t_k,t);\; m\in \mathbb {N},\, (a_k)_{1\le k\le m},\, (t_k)_{1\le k\le m}\subset [0,T] \right\rbrace ,\) 
with respect to the scalar product
\(\langle R_W(t,\cdot ), R_W(s,\cdot )\rangle _{\mathcal {H}_1} = R_W(t,s).\) 
Then, the mapping defined by
\(\mathcal {R}: \mathbb {1}_{[0,t]}\in \mathcal {E}\rightarrow R_W(t,\cdot ) \in \tilde{\mathcal {E}}\) 
can be extended to an isometry \(\mathcal {R}:\mathcal {H}\rightarrow \mathcal {H}_1\)  since
\(\langle \mathcal {R}(\mathbb {1}_{[0,t]}),\mathcal {R}(\mathbb {1}_{[0,s]}) \rangle _{\mathcal {H}_1} = R_W(t,s) = \langle \mathbb {1}_{[0,t]},\mathbb {1}_{[0,s]} \rangle _{\mathcal {H}}.\) 
We have the following integration by parts formula.
Lemme 4.7 For any \(F,G\in S\)  and \(h\in \mathcal {H}\) , we have
\(\mathbb {E}[G \langle DF,h\rangle _{\mathcal {H}}] = -\mathbb {E}[F \langle DG,h\rangle _{\mathcal {H}}] + \mathbb {E}[F GW(h)].\) 

Furthermore, we have the following result.
Proposition 4.8 
For any \(p\ge 1\) , the linear operator \(D\)  is closable from \(L^p(\Omega )\)  to \(L^p(\Omega ; \mathcal {H})\) .

We can iteratively define, for any \(m\ge 1\) ,
\(D^m F = \sum _{k_1,k_2,\ldots ,k_m = 1}^n \partial _{k_1,k_2,\ldots ,k_m}f(W(h_1),\ldots ,W(h_n)) h_{k_1}\otimes \ldots \otimes h_{k_m}\) 
which takes values in \(\mathcal {H}^{\otimes m}\) . Moreover, following Proposition REF , we can close \(D^m\) .
Proposition 4.9 For any \(m\ge 1\)  and \(p\ge 1\) , the linear operator \(D^m\)  is closable from \( S\)  to \(L^p(\Omega ;\mathcal {H}^{\otimes m})\) .

By using the same notation for its extension, the domain of the operator \(D^m\)  is the space \(\mathbb {D}^{m,p}\)  which is the completion of \( S\)  with respect to the norm
\(\Vert F\Vert _{\mathbb {D}^{m,p}} : = \left( \mathbb {E}[|F|^p] + \sum _{k = 1}^m \mathbb {E}[\Vert D^k F\Vert _{\mathcal {H}^{\otimes k}}^p]\right)^{1/p}.\) 
In the previous norm, in the case of multidimensional processes, we remark that \(\Vert \cdot \Vert _{\mathcal {H}^{\otimes k}}\)  is not an operator norm but the Hilbert-Schmidt norm. We also denote by \(\mathbb {D}^{m,p}(\mathcal {H})\)  the \(\mathcal {H}\) -valued random variables that belong in \(\mathbb {D}^{m,p}\) . Furthermore, for any \(k,p\ge 1\) , we have Hölder's inequality
\(\Vert FG\Vert _{\mathbb {D}^{k,p}}\le \Vert F\Vert _{\mathbb {D}^{k,r}}\Vert G\Vert _{\mathbb {D}^{k,q}},\) 
for any \(r,q\ge 1\)  such that \(1/r+1/q = 1/p\) .
We now state a chain rule with respect to the Malliavin derivative.
Proposition 4.10 
Let \(g\in C^1(\mathbb {R}^d;\mathbb {R})\)  with bounded derivatives, \(p\ge 1\)  and \(F = (F^{1},\ldots ,F^{n})\)  be a random vector such that \(F^k\in \mathbb {D}^{1,p}\)  for any \(k\in \lbrace 1,\ldots ,n\rbrace \) . Then, \(g(F)\in \mathbb {D}^{1,p}\)  and
\(D(g(F)) = \sum _{k = 1}^n \partial _k g(F) DF^{k}.\) 


The divergence operator
The divergence operator \(\delta \)  is the adjoint of the derivative \(D\) . In fact, \(\delta \)  is an unbounded operator from \(L^2(\Omega ;\mathcal {H}) = \mathbb {D}^{0,2}(\mathcal {H})\)  to \(L^2(\Omega )\)  such that:

its domain \(\textrm {Dom}(\delta )\)  is the set of random variables \(u\in L^2(\Omega ;\mathcal {H})\)  such that, for all \(F\in \mathbb {D}^{1,2}\) ,
\(\left|\mathbb {E}\left[\langle DF, u \rangle _{\mathcal {H}}\right] \right| \le c_u\Vert F\Vert _{L^2(\Omega )},\) 

for any \(u\in \textrm {Dom}(\delta )\) , we have \(\delta (u)\in L^2(\Omega )\)  and the following duality relation holds, for all \(F\in \mathbb {D}^{1,2}\) ,
\(\mathbb {E}\left[\langle DF, u \rangle _{\mathcal {H}}\right] = \mathbb {E}\left[F \delta (u)\right].\) 

We have the following result concerning the continuity of \(\delta \) .
Theorem 4.11 
For any \(p>1\)  and \(m\ge 1\) , the operator \(\delta \)  is continuous from \(\mathbb {D}^{m-1,p}(\mathcal {H})\)  to \(\mathbb {D}^{m,p}\) . That is, the following inequality holds for any \(u\in \mathbb {D}^{m,p}(\mathcal {H})\) ,
\(\Vert \delta (u)\Vert _{\mathbb {D}^{m-1,p}} \le c_{m,p} \Vert u\Vert _{\mathbb {D}^{m,p}(\mathcal {H})}.\) 

This yields in particular that \(\mathbb {D}^{m,p}(\mathcal {H})\subset \textrm {Dom}(\delta )\)  for any \(p>1\)  and \(m\ge 1\) .
For any random variable \(F\in \mathbb {D}^{1,2}\) , we know that \(DF\)  is a stochastic process in \(\mathcal {H}\)  that we can denote \((D_tF)_{t\in [0,1]}\)  which is defined almost surely with respect to the measure \(\lambda \times \mathbb {P}\)  (where \(\lambda \)  is the usual Lebesgue measure). With this, we can deduce a local property of the derivative operator.
Lemme 4.12 
Let \(F \in \mathbb {D}^{1,2}\cap L^2(\Omega ,\mathcal {F}_{[a,b]},\mathbb {P})\) . Then we have
\(D_s F(\omega ) = 0,\) 
for \((\lambda \times \mathbb {P})\) -almost every \((s,\omega )\in [a,b]^c\times \Omega \) .

We use \( S_b([a,b])\)  the natural restriction of \( S_b = S_b([0,1])\)  to \([a,b]\) . It is dense in \(\mathbb {D}^{1,2}\cap L^2(\Omega ,\mathcal {F}_{[a,b]},\mathbb {P})\) . Then, for any \(F\in S([a,b])\) ,
\(D_sF = \sum _{k = 1}^n \partial _k f(W(h_1),\ldots ,W(h_n)) h_k(s),\) 
where \(h_k\in \mathcal {H}([a,b])\)  which is such that \(h_k(s) = 0\)  for \(s\notin [a,b]\) .

Conditional Integration by parts formula
We can now introduce the Skorokhod integral (which is essentially the divergence operator). For any \([a,b]\subset [0,1]\) , we denote \(\delta _{[a,b]}\)  the Skorokhod integral on \([a,b]\)  which is defined by
\(\delta _{[a,b]}(u) = \delta (u\mathbb {1}_{[a,b]}).\) 
That is, \(\delta _{[a,b]}\)  is an unbounded operator from \(L^2(\Omega ;\mathcal {H})\)  to \(L^2(\Omega )\)  such that, for any \(u\in \textrm {Dom}(\delta )\) , we have, for all \(F\in \mathbb {D}^{1,2}\) ,
\(\mathbb {E}\left[\langle DF, u \rangle _{\mathcal {H}([a,b])}\right] = \mathbb {E}\left[\langle DF, u\mathbb {1}_{[a,b]} \rangle _{\mathcal {H}}\right] = \mathbb {E}\left[F \delta _{[a,b]}(u)\right].\) 
Furthermore,
for any \(G_{[a,b]^c}\in L^2(\Omega ,\mathcal {F}_{[a,b]^c},\mathbb {P}) \cap \mathbb {D}^{1,2}\) , we can also see that, thanks to Lemma REF ,
\(\mathbb {E}\left[ \mathbb {E}\left[\langle DF, u \rangle _{\mathcal {H}([a,b])} |\mathcal {F}_{[a,b]^c}\right]G_{[a,b]^c}\right] & =\mathbb {E}\left[\langle F,u\mathbb {1}_{[a,b]}\rangle _{\mathcal {H}} G_{[a,b]^c}\right]\\&= -\mathbb {E}\left[F \langle DG_{[a,b]^c},u\mathbb {1}_{[a,c]}\rangle _{\mathcal {H}} \right] + \mathbb {E}\left[\langle D(FG_{[a,b]^c}),u\mathbb {1}_{[a,b]}\rangle _{\mathcal {H}}\right]\\&=\mathbb {E}[FG_{[a,b]^c} \delta (u \mathbb {1}_{[a,b]})] = \mathbb {E}\left[FG_{[a,b]^c}\delta _{[a,b]}(u)\right]\\& = \mathbb {E}\left[\mathbb {E}\left[F\delta _{[a,b]}(u)|\mathcal {F}_{[a,b]^c}\right]G_{[a,b]^c}\right].\) 

This yields in fact to the following duality relation under a conditional expectation since \(\mathbb {D}^{1,2}\)  is dense in \(L^2(\Omega )\) ,
\(\mathbb {E}\left[\langle DF, u \rangle _{\mathcal {H}([a,b])} |\mathcal {F}_{[a,b]^c}\right] = \mathbb {E}\left[F\delta _{[a,b]}(u)|\mathcal {F}_{[a,b]^c}\right].\) 
We can now proceed to prove a conditional integration by parts formula.
Proposition 4.13 Let \(F\in \mathbb {D}^{1,2}\) , \(G\)  be a random variable and \(u\)  be an \(L^2([a,b])\) -valued random variable such that
\(\langle DF,u\rangle _{\mathcal {H}([a,b])} \ne 0\quad \textrm {and}\quad G \frac{u}{ \langle DF,u\rangle _{\mathcal {H}([a,b])}} \in \textrm {Dom}(\delta ).\) 
Then, for any function \(f\in C^1\)  with bounded derivatives, we have that
\(\mathbb {E}\left[f^{\prime }(F)G | \mathcal {F}_{[a,b]^c}\right] = \mathbb {E}\left[f(F)H_{[a,b]}(F,G)| \mathcal {F}_{[a,b]^c}\right],\) 
where
\(H_{[a,b]}(F,G) : = \delta _{[a,b]}\left(G \frac{u}{ \langle DF,u\rangle _{\mathcal {H}([a,b])}}\right).\) 

We apply Proposition REF  to deduce that
\(f^{\prime }(F) = \left\langle D(f(F)) ,\frac{u}{ \langle DF , u \rangle _{\mathcal {H}([a,b])} }\right\rangle _{\mathcal {H}([a,b])}.\) 
Then, it follows from (REF ) that
\(\mathbb {E}\left[ f^{\prime }(F) G|\mathcal {F}_{[a,b]^c}\right] &= \mathbb {E}\left[ \langle D(f(F)),u\rangle _{\mathcal {H}([a,b])} (\langle D F,u\rangle _{\mathcal {H}([a,b])})^{-1}G|\mathcal {F}_{[a,b]^c}\right]\\ &= \mathbb {E}\left[ \langle D(f(F)),Gu(\langle D F,u\rangle _{\mathcal {H}([a,b])})^{-1} \rangle _{\mathcal {H}([a,b])} |\mathcal {F}_{[a,b]^c}\right]\\ &= \mathbb {E}\left[ f(F)\delta _{[a,b]}\left(Gu(\langle D F,u\rangle _{\mathcal {H}([a,b])})^{-1} \right)|\mathcal {F}_{[a,b]^c}\right],\) 

which is the desired result.
We may now introduce the Malliavin matrix associated to a random vector.
Definition 4.14 
Let \(p>1\)  and \(F = (F^1,\ldots ,F^m)\)  be a random vector such that \(F^k\in \mathbb {D}^{1,p}\)  for all \(k\in \lbrace 1,\ldots ,m\rbrace \) . The covariance matrix \(\gamma _F\)  associated to \(F\)  is defined as
\(\gamma _F : = (\langle DF^i,DF^j \rangle _{\mathcal {H}})_{(i,j)\in \lbrace 1,\ldots ,m\rbrace ^2}.\) 

In the case \(\mathcal {H}= \mathcal {H}([a,b])\) , we use the notation \(\gamma _{F,[a,b]}\) . We then have the following Lemma [6]} :
Lemme 4.15 
Let \(\gamma \)  be a \(m\times m\)  random matrix such that \(\textrm {det}(\gamma )>0\)  almost surely and \(\textrm {det}(\gamma )^{-1}\in \cap _{p\ge 1}L^p(\Omega )\) . We assume that \((\gamma )_{i,j}\in \mathbb {D}^{\infty } : = \cap _{p\ge 1}\cap _{k\ge 1} \mathbb {D}^{k,p}\)  for any \((i,j)\in \lbrace 1,\ldots ,m\rbrace ^2\) . Then, \((\gamma ^{-1})_{i,j}\in \mathbb {D}^{\infty }\) , for any \((i,j)\in \lbrace 1,\ldots ,m\rbrace ^2\) , and
\(D(\gamma ^{-1})_{i,j} = -\sum _{k,\ell = 1}^m (\gamma ^{-1})_{i,k}(\gamma ^{-1})_{\ell ,j} D\gamma _{k,\ell }.\) 

Thanks to the Malliavin matrix, we define a nondegeneracy condition for random vectors.
Definition 4.16 We say that a random vector \(F = (F^1,\ldots ,F^m)\) , whose components are in \(\mathbb {D}^{\infty }\) , is nondegenerate if the Malliavin matrix \(\gamma _F\)  is invertible almost surely and
\(\textrm {det}(\gamma _F)^{-1}\in \cap _{p\ge 1}L^p(\Omega ).\) 

We now state a more general conditional integration by parts result for nondegenerate random vectors.
Proposition 4.17 
Let \(F = (F^1,\ldots ,F^m)\)  be a nondegenerate random vector, \(G\in \mathbb {D}^{\infty }\)  and \(g\in C^{\infty }_p(\mathbb {R}^m)\) . Then, for any multiindex \(\alpha \in \lbrace 1,\ldots ,m\rbrace ^k\) , \(k\ge 1\) , there exists an element \(H_{\alpha ,[a,b]}(F,G)\in \mathbb {D}^{\infty }\)  such that
\(\mathbb {E}\left[ \partial ^{\alpha }g(F) G|\mathcal {F}_{[a,b]^c}\right] = \mathbb {E}\left[ g(F) H_{\alpha ,[a,b]}(F,G)|\mathcal {F}_{[a,b]^c}\right].\) 
The variable \(H_{\alpha ,[a,b]}(F,G)\)  is given recursively by
\(H_{\alpha ,[a,b]}(F,G) = H_{\alpha _k,[a,b]}(F,H_{(\alpha _1,\ldots ,\alpha _{m-1}),[a,b]}(F,G))\) 
where, for any \(i\in \lbrace 1,\ldots ,m\rbrace \) ,
\(H_{i,[a,b]}(F,G) = \sum _{j = 1}^m\delta _{[a,b]}\left(G (\gamma _{F,[a,b]}^{-1})_{i,j} DF^j \right).\) 

It follows from the chain rule that, for any \(j\in \lbrace 1,\ldots ,m\rbrace \) ,
\(\langle D(g(F)),DF^j \rangle _{\mathcal {H}([a,b])} = \sum _{i = 1}^m \partial _i g(F) \langle DF^i,DF^j\rangle _{\mathcal {H}([a,b])} = \sum _{i = 1}^m \partial _i g(F) (\gamma _{F,[a,b]})_{i,j}.\) 
Thus, we obtain that
\(\partial _i g(F) = \sum _{j = 1}^m (\gamma _{F,[a,b]}^{-1})_{i,j} \langle D(g(F)),DF^j \rangle _{\mathcal {H}([a,b])}.\) 
By exploiting the duality relation (REF ), this gives
\(\mathbb {E}\left[ \partial _ig(F) G|\mathcal {F}_{[a,b]^c}\right] &= \mathbb {E}\left[ \left\langle D(g(F)),\sum _{j = 1}^mG(\gamma _{F,[a,b]}^{-1})_{i,j} DF^j \right\rangle _{\mathcal {H}([a,b])}|\mathcal {F}_{[a,b]^c}\right]\\&= \mathbb {E}\left[ g(F) H_{i,[a,b]}(F,G)|\mathcal {F}_{[a,b]^c}\right].\) 

Thanks to Lemma REF , we can deduce that \(\gamma _{F,[a,b]}\in \mathbb {D}^{\infty }\) . Furthermore, by Theorem REF , this yields that \(H_{i,[a,b]}(F,G)\in \mathbb {D}^{\infty }\)  if \(G\in \mathbb {D}^{\infty }\) . From here, we can proceed by induction and deduce the result.
We finally give an estimate for the random variables \(H_{\alpha ,[a,b]}\) .
Lemme 4.18 
Let \(k\ge 1\)  and \(p>1\) . For any \(\alpha \in \lbrace 1,\ldots ,m\rbrace ^k\) , we have
\(\Vert H_{\alpha ,[a,b]}(F,G)\Vert _{L^p(\Omega )} \le \Vert G\Vert _{\mathbb {D}^{k,q}} \prod _{\ell = 1}^{k} \left\Vert \sum _{j =1}^m (\gamma _{F,[a,b]}^{-1})_{\alpha _\ell ,j}DF^j \right\Vert _{\mathbb {D}^{k-\ell +1,r_\ell }}\) 
where \(1/p = 1/q + \sum _{\ell = 1}^m 1/r_\ell \) .

By Theorem REF  and Hölder's inequality, we have
\(\Vert H_{\alpha ,[a,b]}(F,G)\Vert _{L^p(\Omega )} &\le c_{p,m}\left\Vert H_{(\alpha _1,\ldots ,\alpha _{k-1}),[a,b]}(F,G) \sum _{j =1}^m (\gamma _{F,[a,b]}^{-1})_{\alpha _k,j}DF^j \right\Vert _{\mathbb {D}^{1,p}}\\ &\le c_{p,m}\left\Vert H_{(\alpha _1,\ldots ,\alpha _{k-1}),[a,b]}(F,G)\right\Vert _{\mathbb {D}^{1,q}}\left\Vert \sum _{j =1}^m (\gamma _{F,[a,b]}^{-1})_{\alpha _k,j}DF^j \right\Vert _{\mathbb {D}^{1,r}},\) 

where \(1/p = 1/q + 1/r\) , which yields the desired result by induction.

Malliavin calculus and rough paths
The idea of this Section is to use Malliavin calculus (and especially the conditional bounds proved in the previous subsection) to derive some nice bounds on flow generated by rough differential equations. In order to do so, we will need some properties of the Cameron-Martin spaces \(\mathcal {H}_1\)
Embedding of the spaces \(\mathcal {H}\)  and \(\mathcal {H}_1\)
In this section, we give some embedding of \(\mathcal {H}\)  and \(\mathcal {H}_1\)  into the space of continuous functions with finite \(p\) -variation, with \(p>0\) .
Definition 5.1 Let \(f:[0,T]\rightarrow \mathbb {R}^d\)  be a continuous function. For any \(p>0\) , we define its \(p\) -variation on \([a,b]\subset [0,T]\)  as
\(\Vert f\Vert _{p\textrm {-var};[a,b]} = \sup _{\pi \in \Pi ([a,b])}\left( \sum _{[u,v]\in \pi } |f(v) - f(u)|^p\right)^{1/p},\) 
where we remind that \(\Pi ([a,b])\)  is the set of all subdivisions of \([a,b]\) . The set of continuous functions with finite \(p\) -variation on \([a,b]\)  is denoted \(\mathcal {C}^{p\textrm {-var}}([a,b];\mathbb {R}^d) = \mathcal {C}^{p\textrm {-var}}([a,b])\) .

A, similar, mixed \((p,q)\) -variation notion also exists for continuous functions on \([0,T]^2\) .
Definition 5.2 Let \(R:[0,T]^2\rightarrow \mathbb {R}^d\)  be a continuous function. For any \(p,q>0\) , we define its \((p,q)\) -variation on the square \([a,b]\times [c,d]\)  as
\(\Vert R\Vert _{(p,q)\textrm {-var};[a,b]\times [c,d]} = \sup _{\begin{array}{c}\pi _1\in \Pi ([a,b])\\\pi _2\in \Pi ([c,d])\end{array}}\left( \sum _{[u_2,v_2]\in \pi _1} \left(\sum _{[u_1,v_1]\in \pi _2}|\square _{[u_1,v_1]\times [u_2,v_2]} R|^p\right)^{q/p}\right)^{1/q},\) 
where \(\square _{[u_1,v_1]\times [u_2,v_2]} R = R(v_1,v_2) - R(v_1,u_2) - R(u_1,v_2) + R(u_1,u_2)\) . The set of continuous functions with finite \((p,q)\) -variation on \([a,b]\times [c,d]\)  is denoted \(\mathcal {C}^{(p,q)\textrm {-var}}([a,b]\times [c,d];\mathbb {R}^d) = \mathcal {C}^{(p,q)\textrm {-var}}([a,b]\times [c,d])\) .

We also denote \(\Vert R\Vert _{p\textrm {-var};[a,b]} = \Vert R\Vert _{(p,p)\textrm {-var};[a,b]^2}\)  as well as \( \mathcal {C}^{p\textrm {-var}}([a,b]^2) : = \mathcal {C}^{(p,p)\textrm {-var}}([a,b]^2)\) .
Remark 5.3 We can see that
\(\Vert R\Vert _{p\vee q\textrm {-var};[a,b]} \le \Vert R\Vert _{(p,q)\textrm {-var};[a,b]^2} \le \Vert R\Vert _{p\wedge q\textrm {-var};[a,b]}.\) 

Furthermore, we have the following definition.
Definition 5.4 Let \(\rho \in [1,2)\)  and \([a,b]\subset [0,T]\) . We say that \(R\)  has finite Hölder-controlled \(\rho \) -variation on \([a,b]\)  if \(\Vert R\Vert _{\rho \textrm {-var};[a,b]}<+\infty \)  and if the following estimate holds, for any \([s,t]\subset [a,b]\) ,
\(\Vert R\Vert _{\rho \textrm {-var};[s,t]} \lesssim (t-s)^{1/\rho }.\) 

We are now in position to state a first assumption on the process \(W\)  that we will consider on the rest of the section.
Assumption 5.5 
The process \(W\)  is a \(\mathbb {R}^d\) -valued continuous centered Gaussian process starting at 0 with iid components and covariance \(R_W\)  which belongs in \(\mathcal {C}^{(1,\rho )\textrm {-var}}([0,T]^2)\)  for some \(\rho \in [1,2)\) . Furthermore, the following estimate holds, for any \([a,b]\subset [0,T]\) ,
\(\Vert R_W\Vert _{(1,\rho )\textrm {-var};[a,b]^2} \lesssim (b-a)^{1/\rho }.\) 

We immediately see that (REF ) is verified if the covariance has finite Hölder-controlled \(\rho \) -variation on \([0,T]\) . This leads in particular to the following hypothesis.
Assumption 5.6 
The process \(W\)  is a \(\mathbb {R}^d\) -valued continuous centered Gaussian process starting at 0 with iid components and covariance \(R_W\)  which has finite Hölder-controlled \(\rho \) -variation on \([0,T]\)  for some \(\rho \in [1,2)\) .

In the following, we denote
\(\kappa _{a,b} : = \Vert R_W\Vert _{(1,\rho )\textrm {-var};[a,b]^2}^{1/2}\) 
and
\(\sigma _{a,b} : = \mathbb {E}\left[\left(W^{(j)}_b - W^{(j)}_a \right)^2 \right]^{1/2} = (\square _{[a,b]^2}R_W)^{1/2},\) 
for any \(j\in \lbrace 1,\ldots ,n\rbrace \) .
We can now proceed to state our first embedding :
Theorem 5.7 ([7]}, Theorem 1.1) 
Let \(W\)  be a centered Gaussian process satisfying Assumption REF . Then, for any \(h\in \mathcal {H}_1\) , \([a,b]\subset [0,T]\) , we have
\(\Vert h\Vert _{q\textrm {-var};[a,b]} \le \kappa _{a,b}\Vert h\Vert _{\mathcal {H}_1},\) 
where \(q = 1/(1/2\rho +1/2)<2\) . In particular, the embedding \(\mathcal {H}_1\hookrightarrow \mathcal {C}^{q\textrm {-var}}([a,b])\)  is continuous.

Furthermore, we also have the following results [8]}.
Proposition 5.8 Let \(W\)  be a centered Gaussian process satisfying Assumption REF  and \([a,b]\subset [0,T]\) .

Let \(f\in \mathcal {C}^{p\textrm {-var}}([a,b])\)  with \(1/p+1/\rho >1\) . Then, \(f\in \mathcal {H}\)  and
\(\Vert f\Vert _{\mathcal {H}}^2 = \int _a^b\int _a^b f(s)f(t) \mathop {}\!\mathrm {d}R_W(s,t),\) 
where the right-hand side is well defined as a 2d Young integral. In particular, we have the continuous embedding \(\mathcal {C}^{p\textrm {-var}}([a,b])\hookrightarrow \mathcal {H}([a,b])\)  since
\(\Vert f\Vert _{\mathcal {H}([a,b])}^2 \lesssim \Vert f\Vert _{p\textrm {-var};[a,b]}^2 \Vert R_W\Vert _{\rho \textrm {-var};[a,b]}.\) 

Let \(f_1\in \mathcal {C}^{p\textrm {-var}}([a,b])\)  with \(1/p+1/\rho >1\)  and \(f_2\in \mathcal {H}([a,b])\) . Then,
\(\langle f_1,f_2 \rangle _{\mathcal {H}([a,b])} = \int _a^b f_1\mathop {}\!\mathrm {d}\mathcal {R}f_2,\) 
where the right-hand side is well defined as a Young integral and \(\mathcal {R}: \mathcal {H}\rightarrow \mathcal {H}_1\)  is the isomorphism defined by (REF ).


We now make an additional assumption on the process \(W\) .
Assumption 5.9 
Let \(W\)  be an \(\mathbb {R}^d\) -valued Gaussian process with iid coordinates and covariance function \(R_W\)  such that

it has non-positively correlated increments, that is, for all \((t_1,t_2,t_3,t_4)\in [0,T]^4\)  with \(t_1<t_2<t_3<t_4\) , we have
\(\square _{[t_1,t_2]\times [t_3,t_4]}R_W \le 0,\) 

its covariance function is diagonally dominant, that is, for all \((t_1,t_2,t_3,t_4)\in [0,T]^4\)  with \(t_1<t_2<t_3<t_4\) , we have
\(\square _{[t_2,t_3]\times [t_1,t_4]}R_W \ge 0.\) 


We deduce the following inequalities [8]}.
Proposition 5.10 Let \(W\)  be a Gaussian process satisfying Assumption REF , \(p\ge 1\)  such that \(1/p+1/\rho >1\) .

For every \(f\in \mathcal {C}^{p\textrm {-var}}([a,b])\) , we have
\(\Vert f\Vert _{\mathcal {H}([a,b])}^2\lesssim \kappa _{a,b}^2\left(\Vert f\Vert _{p\textrm {-var};[a,b]}^2 + \Vert f\Vert _{\infty ;[a,b]}^2 \right)\) 

If \(W\)  satisfies Assumption REF , we have, for any \(f\in \mathcal {C}^{\gamma }([a,b])\)  with \(1/\rho +\gamma >1\) ,
\(\Vert f\Vert _{\mathcal {H}([a,b])}^2\ge \sigma _{a,b}^2 \min _{t\in [a,b]} |f(t)|.\) 


To continue, we need some non-degeneracy concerning \(W\) .
Assumption 5.11 
Let \(W\)  be a centered continuous \(\mathbb {R}^d\) -valued Gaussian process. We assume that there exists an \(\alpha >0\)  such that
\(\inf _{0\le s\le t\le T} (t-s)^{-\alpha } \operatorname{Var}\left[W_t-W_s| \mathcal {F}_{[0,s]}\vee \mathcal {F}_{[t,1]}\right] = c_W >0.\) 
The smallest \(\alpha \)  that satisfies the above condition is called the index of non-determinism of \(W\) .

Under the previous assumption, we deduce our last inequality which is taken from [10]}.
Proposition 5.12 
Let \(W\)  be a continuous Gaussian process satisfying Assumptions REF , REF  and REF . Then, for any \(f\in \mathcal {C}^{\gamma }([a,b])\)  with \(\gamma +1/\rho >1\) , we have
\(\Vert f\Vert _{\infty ;[a,b]} \le 2\max \left(\sigma _{a,b}^{-1}\Vert f\Vert _{\mathcal {H}([a,b])}, \sqrt{c_W}^{-1}\Vert f\Vert _{\mathcal {H}([a,b])}^{2\gamma /(2\gamma + \alpha )}\Vert f\Vert _{\mathcal {C}^{\gamma }([a,b])}^{\alpha /(2\gamma +\alpha )} \right).\) 

As a direct consequence, we have the following result.
Corollary 5.13 
Let \(W\)  be a continuous Gaussian process satisfying Assumptions REF , REF  and REF . Then, for any \(f\in \mathcal {C}^{\gamma }([a,b])\)  with \(\gamma +1/\rho >1\) , we have
\(\Vert f\Vert _{\mathcal {H}([a,b])} \ge \frac{\sigma _{a,b} \Vert f\Vert _{\infty ;[a,b]}}{2}\min \left(1,\frac{2(c_W/2)^{(2\gamma +\alpha )/4\gamma }\Vert f\Vert _{\infty ;[a,b]}^{\alpha /2\gamma }}{\sigma _{a,b}|f|_{\mathcal {C}^{\gamma }([a,b])}^{\alpha /2\gamma }} \right).\) 


Gaussian rough paths
Let us now focus ourselves on rough differential equations driven by Gaussian rough paths Let us consider \(W\)  a continuous centered Gaussian process satisfying Assumption REF . It can therefore be lifted to a geometric rough path (see [11]}) that we denote \(\mathbf {W}\) .
Proposition 5.14 
Let \(W\)  be a continuous Gaussian process satisfying
Assumption REF . Then, almost surely, \(W\)  can be lifted to a geometric \(p\) -rough path \(\mathbf {W}\)  with \(p>2\rho \)  and verifies
\(\mathbb {E}\left[\exp \left(\eta \Vert \mathbf {W}\Vert _{1/p\textrm {-var};[0,T]}\right)\right], \quad \textrm {for some}\;\eta >0.\) 

We also have the following useful result (which is a direct consequence of the Proposition REF  and [11]}).
Proposition 5.15 
Let \(W\)  be a Gaussian process satisfying Assumption REF . Then, \(W\)  can be lifted to a geometric \(p\) -rough path \(\mathbf {W}\)  with \(p>2\rho \)  which has \(1/p\) -Hölder sample paths and verifies
\(\mathbb {E}\left[\exp \left(\eta \Vert \mathbf {W}\Vert _{1/p\textrm {-Höl};[0,T]}\right)\right], \quad \textrm {for some}\;\eta >0.\) 

We will need the following more precise statement of Theorem (REF ) in this setting :
Theorem 5.16 
Let \(p\ge 1\) , \(\mathbf {w}\)  be a weakly geometric \(p\) -rough path, \(\sigma \in \mathcal {C}^{\gamma }_b(\mathbb {R}^d;\mathbb {R}^{d\times d})\)  for some \(\gamma >p\) . Let \(a\in [0,T)\)  and \(x\in \mathbb {R}^d\) .
There exists a unique \(p\) -weakly geometric rough path \(\mathbf {x}\in \mathcal {C}^{p-var}([a,T]; G^{\lfloor p \rfloor }(\mathbb {R}^d))\)  such that
\(\mathbf {x}_{a,\cdot }^{1} = \varphi _{a,\cdot }(x)\) 
where \(\varphi \)  is the flow constructed in Theorem REF .
Furthermore, if
\((w^{\varepsilon })_{\varepsilon >0}\)  is a family of smooth functions on \([0,T]\)  such that
\(S_{\lfloor p\rfloor }(w^{\varepsilon })\rightarrow \mathbf {w},\) 
in \(p\) -variation.
Then, for any \(0\le a\le t\le T\) , the solutions \((x^{\varepsilon })_{\varepsilon >0}\subset \mathcal {C}^{1}([s,T])\)  of
\(x_t^{x,a,\varepsilon } = x + \sum _{k = 1}^d \int _a^t \sigma _k(x_r^{x,a,\varepsilon }) \mathop {}\!\mathrm {d}w^{k,\varepsilon }_r,\) 
are such that
\(S_{\lfloor p\rfloor }(x^{\varepsilon }) \rightarrow \mathbf {x},\) 
in \(p\) -variation, where \(\mathbf {x}\) . Moreover, for any \([s,t]\subset [a,T]\) , the following estimates hold
\(\Vert \mathbf {x}\Vert _{p\textrm {-var};[s,t]} \lesssim _{p,\gamma } \left(\Vert \sigma \Vert _ {\mathcal {C}^{\gamma -1}_b}\Vert \mathbf {w}\Vert _{p\textrm {-var};[s,t]} \vee \Vert \sigma \Vert _ {\mathcal {C}^{\gamma -1}_b}^p\Vert \mathbf {w}\Vert _{p\textrm {-var};[s,t]}^p\right).\) 

As a direct consequence of the previous result, we deduce.
Corollary 5.17 Let \(W\)  be a Gaussian process satisfying Assumption REF  and let \(p>2\rho \) . Let \(\mathbf {X}\)  be the unique solution of equation
\(\mathop {}\!\mathrm {d}X_t = \sigma (X_t) \mathop {}\!\mathrm {d}\mathbf {W}_t,\quad X_a = x,\quad t\in [a,T], \) 
in the sense of Theorem REF . Let us denote \(X^{x,a}_{t}= \mathbf {X}^{1}_{a,t}\) .
Then, for any \([s,t]\subset [a,T]\) ,
\(\sup _{x\in \mathbb {R}^d}|X^{x,a}-x|_{\mathcal {C}^{\frac{1}{p}}([s,t])} \lesssim _{\alpha ,\gamma } \left(\Vert \sigma \Vert _ {\mathcal {C}^{\gamma -1}_b}\Vert \mathbf {W}\Vert _{p\textrm {-var};[s,t]} \vee \Vert \sigma \Vert _ {\mathcal {C}^{\gamma -1}_b}^p\Vert \mathbf {W}\Vert _{p\textrm {-var};[s,t]}^p\right).\) 


Malliavin calculus on rough differential equations
The solution \(X\)  of a RDE driven by a Gaussian process \(W\)  is a random variable inheriting its randomness from \(W\) .
One can thus try to differentiate \(X\)  in the Malliavin sense. The Malliavin derivative of \(X\)  can indeed be simply expressed thanks to the Jacobian \(\mathbf {J}\)  of the solution of equation (REF ). It is given by \(\left(\mathbf {J}^{x,a}_{t}\right)_{(i,j)\in \lbrace 1,\ldots ,d\rbrace ^2} = \partial _{x_j} (X_t^{x,a})_{i}\)  and solution the following linear RDE
\(\mathbf {J}_t^{x,a} = \mathrm {Id} + \sum _{k = 1}^d \int _a^t \nabla \sigma _k(X_s^{x,a}) \mathbf {J}_s^{x,a} \mathop {}\!\mathrm {d}\mathbf {W}_s^k,\) 
where \(\sigma _k\)  is the \(k\) -th column of \(\sigma \) .
We notice that the inverse Jacobian \(\mathbf {J}^{-1}\)  solves
\((\mathbf {J}^{x,a}_t)^{-1} = \mathrm {Id} - \sum _{k = 1}^d \int _a^t (\mathbf {J}^{x,a}_s)^{-1} \nabla \sigma _k(X_s^{x,a}) \mathop {}\!\mathrm {d}\mathbf {W}_s^k,\) 
and that, for any \(s\le r\le t\) , we have, by the flow property of the Jacobian,
\(\mathbf {J}_t^{x,r} = \mathbf {J}_t^{x,s}(\mathbf {J}^{x,s}_r)^{-1}.\) 
We have the following results concernant \(\mathbf {J}\)  as well as the Malliavin derivative of \(X\)  [13]}, [14]}, [15]}.
Proposition 5.18 Let \(W\)  be a continuous centered Gaussian process satisfying Assumption REF  and \(\sigma \in \mathcal {C}^{\infty }_b(\mathbb {R}^d;\mathbb {R}^{d\times d})\) .

For any \(q\ge 1\) , there exists a constant \(c_{q,a}\)  such that the Jacobian \(\mathbf {J}\)  defined by (REF ) satisfies
\(\mathbb {E}\left[ \sup _{x\in \mathbb {R}^d}\Vert \mathbf {J}^{x,a}\Vert _{p\textrm {-var};[0,T]}^q\right] = c_{q,a}.\) 

For any \(x\in \mathbb {R}^d\)  and \(t>0\) , we have that \(X_t\in \mathbb {D}^{\infty }\)  and, furthermore, the Malliavin derivative \(D_sX_t^{x,a}\)  is such that
\(D_sX_t^{x,a} = \mathbf {J}_{t}^{x,s}\sigma (X_s^{x,a}),\) 
for any \(a\le s\le t\) , and \(D_sX_t^{x,a} = 0\)  for all \(s>t\) .


Remark 5.19 We remark that Equation (REF ) can also be replaced by
\(\mathbb {E}\left[ \sup _{x\in \mathbb {R}^d}\Vert \mathbf {J}^{x,a}\Vert _{1/p\textrm {-Höl};[0,T]}^q\right] = c_{q,a}.\) 

We now state some estimates on the Malliavin derivative of \(X\)  as well as the associated covariance matrix \(\gamma _X\) . We follow the approach of [15]} which uses some functionals \(\Xi _m\)  that are equal to \(D^m_h X_t\)  along a single path \(h\in \mathcal {H}([a,b])\)  (the identification extends to any direction \((h_1,h_2,\ldots ,h_m)\)  since these are bounded symmetric \(m\) -multilinear maps). For any \(m\ge 2\) , the Malliavin derivative of \(X\)  along \(h^{\otimes m}\in \mathcal {H}^{\otimes m}([a,b])\)  is given by
\(D^m_h X_t^{x,a} &: = \langle D^m X_t^{x,a}, h^{\otimes m} \rangle _{\mathcal {H}^{\otimes m}([a,b])}\\ & =\sum _{k = 1}^d\sum _{\ell = 2}^m\sum _{\begin{array}{c}\mathbf {i}\in K_{\ell }\\ |\mathbf {i}| = m\end{array}} C_{1,\mathbf {i}}\int _a^t \mathbf {J}_{t}^{x,s} \nabla _{D^{i_1}_hX_s^{x,a}, D^{i_2}_h X_s^{x,a}, \ldots ,D^{i_\ell }_h X_s^{x,a}}^{\ell } \sigma _k(X_s^{x,a})\mathop {}\!\mathrm {d}\mathbf {W}^k_s\\ &\hspace{10.0pt} + \sum _{k = 1}^d\sum _{\ell = 1}^{m-1} \sum _{\begin{array}{c}\mathbf {i}\in K_{\ell } \\ |\mathbf {i}| = m\end{array}} C_{2,\mathbf {i}}\int _a^t \mathbf {J}_{t}^{x,s} \nabla _{D^{i_1}_hX_s^{x,a}, D^{i_2}_h X_s^{x,a}, \ldots ,D^{i_\ell }_h X_s^{x,a}}^{\ell } \sigma _k(X_s^{x,a}) \mathop {}\!\mathrm {d}\mathbf {h}_s^k\) 

where \(K_{\ell } = \lbrace \mathbf {j}\in \mathbb {N}^{\ell };\; 0<j_1\le j_2\le \ldots \le j_{\ell }\rbrace \) , for some constants \(\lbrace C_{1,\mathbf {i}}\rbrace _{\mathbf {i}\in \mathbf {K}_{\ell },2\le \ell \le m}\)  and \(\lbrace C_{2,\mathbf {i}}\rbrace _{\mathbf {i}\in \mathbf {K}_{\ell },1\le \ell \le m-1}\) . Furthermore, by Equation (REF ), Leibniz's differentiation rule and Faà di Bruno's formula, the Malliavin derivative of \((\mathbf {J})^{-1}\)  along \(h^{\otimes m}\in \mathcal {H}^{\otimes m}([a,b])\)  is such that
\(&D_h^m(\mathbf {J}^{x,a}_t)^{-1} =\\ &\hspace{5.0pt} -\sum _{k = 1}^d\sum _{\ell = 1}^{m} \sum _{o = 1}^{\ell }\sum _{\begin{array}{c}\mathbf {i}\in K_{o}\\ |\mathbf {i}| = \ell \end{array}} C_{3,m,\ell ,\mathbf {i}}\int _a^t(\mathbf {J}^{x,s}_t)^{-1} D_h^{m-\ell }(\mathbf {J}^{x,a}_s)^{-1} \nabla ^{o}_{D^{i_1}_hX_s^{x,a}, \ldots ,D^{i_o}_h X_s^{x,a}}(\nabla \sigma _k(X_s^{x,a})) \mathop {}\!\mathrm {d}\mathbf {W}_s^k\\ &\hspace{5.0pt}-\sum _{k = 1}^d\sum _{\ell = 1}^{m-1} \sum _{o = 1}^{\ell } \sum _{\begin{array}{c}\mathbf {i}\in K_{o}\\ |\mathbf {i}| = \ell \end{array}} C_{4,m,\ell ,\mathbf {i}}\int _a^t (\mathbf {J}^{x,s}_t)^{-1} D_h^{m-1-\ell }(\mathbf {J}^{x,a}_s)^{-1} \nabla ^{o}_{D^{i_1}_hX_s^{x,a}, \ldots ,D^{i_o}_h X_s^{x,a}}(\nabla \sigma _k(X_s^{x,a})) \mathop {}\!\mathrm {d}\mathbf {h}_s^k\\ &\hspace{5.0pt}-m\sum _{k = 1}^d\int _a^t(\mathbf {J}^{x,s}_t)^{-1} D_h^{m-1}(\mathbf {J}^{x,a}_s)^{-1} \nabla \sigma _k(X_s^{x,a}) \mathop {}\!\mathrm {d}\mathbf {h}_s^k,\) 

for some constants \(\lbrace C_{3,m,\ell ,\mathbf {i}}\rbrace _{\begin{array}{c}\mathbf {i}\in K_{o},1\le o\le \ell \\ 1\le \ell \le m\end{array}}\)  and \(\lbrace C_{4,m,\ell ,\mathbf {i}}\rbrace _{\begin{array}{c}\mathbf {i}\in K_{o},1\le o\le \ell \\ 1\le \ell \le m\end{array}}\) .
Proposition 5.20 Let \(B\)  be an independent copy of \(W\)  and let \(\mathbf {B}\)  the correspondgin rough path above \(B\) .
For any \(t\in [a,b]\) , we denote by
\(\mathcal {X}_{1,t}(\mathbf {W},\mathbf {B}) = \sum _{k = 1}^d\int _a^t \mathbf {J}_{t}^{x,s}\sigma _k(X_s^{x,a}) \mathop {}\!\mathrm {d}\mathbf {B}^k_s,\) 
\(\mathcal {Y}_{1,t}(\mathbf {W},\mathbf {B}) = - \sum _{k = 1}^d \left(\int _a^t (\mathbf {J}_{t}^{x,a})^{-1}\nabla _{\mathcal {X}_{1,s}(\mathbf {W},\mathbf {B})}(\nabla \sigma _k(X_s^{x,a}))\mathop {}\!\mathrm {d}\mathbf {W}^k_s - \int _a^t (\mathbf {J}_{t}^{x,a})^{-1}\nabla \sigma _k(X_s^{x,a})\mathop {}\!\mathrm {d}\mathbf {B}^k_s\right),\) 
and, for any \(m\ge 2\) ,
\(\mathcal {X}_{m,t}(\mathbf {W},\mathbf {B}) &= \sum _{k = 1}^d\left( \sum _{\ell = 2}^m \sum _{\begin{array}{c}\mathbf {i}\in \mathbf {K}_{\ell } \\ |\mathbf {i}| = m\end{array}} C_{1,\mathbf {i}} \int _a^t \mathbf {J}_{t}^{x,s} \nabla _{\mathcal {X}_{i_1,s}(\mathbf {W},\mathbf {b}), \ldots ,\mathcal {X}_{i_\ell ,s}(\mathbf {W},\mathbf {b})}^{\ell } \sigma _k(X_s^{x,a}) \mathop {}\!\mathrm {d}\mathbf {W}^k_s\right.\nonumber \\ &\hspace{10.0pt} +\left.\sum _{\ell = 1}^{m-1} \sum _{\begin{array}{c}\mathbf {i}\in \mathbf {K}_{\ell } \\ |\mathbf {i}| = m\end{array}} C_{2,\mathbf {i}} \int _a^t \mathbf {J}_{t}^{x,s} \nabla _{\mathcal {X}_{i_1,s}(\mathbf {W},\mathbf {b}), \ldots ,\mathcal {X}_{i_\ell ,s}(\mathbf {W},\mathbf {b})}^{\ell } \sigma _k(X_s^{x,a}) \mathop {}\!\mathrm {d}\mathbf {B}_s^k\right),\) 
\(&\mathcal {Y}_{m,t}(\mathbf {W},\mathbf {B}) =\nonumber \\ & -\sum _{k = 1}^d\left( \sum _{\ell = 1}^{m} \sum _{o = 1}^{\ell }\sum _{\begin{array}{c}\mathbf {i}\in \mathbf {K}_{o}\\ |\mathbf {i}| = \ell \end{array}} C_{3,m,\ell ,\mathbf {i}} \int _a^t(\mathbf {J}^{x,s}_t)^{-1} \mathcal {Y}_{m-\ell ,s}(\mathbf {W},\mathbf {B}) \nabla ^{o}_{\mathcal {X}_{i_1,s}(\mathbf {W},\mathbf {B}), \ldots ,\mathcal {X}_{i_o,s}(\mathbf {W},\mathbf {B})}(\nabla \sigma _k(X_s^{x,a})) \mathop {}\!\mathrm {d}\mathbf {W}_s^k\right.\nonumber \\ &\hspace{10.0pt}+\sum _{\ell = 1}^{m-1} \sum _{o = 1}^{\ell } \sum _{\begin{array}{c}\mathbf {i}\in \mathbf {K}_{o}\\ |\mathbf {i}| = \ell \end{array}} C_{4,m,\ell ,\mathbf {i}}\int _a^t (\mathbf {J}^{x,s}_t)^{-1} \mathcal {Y}_{m-1-\ell ,s}(\mathbf {W},\mathbf {B}) \nabla _{\mathcal {X}_{i_1,s}(\mathbf {W},\mathbf {B}), \ldots ,\mathcal {X}_{i_o,s}(\mathbf {W},\mathbf {B})}^{o}(\nabla \sigma _k(X_s^{x,a})) \mathop {}\!\mathrm {d}\mathbf {B}_s^k \nonumber \\ &\hspace{10.0pt} +\left.m\int _a^t(\mathbf {J}^{x,s}_t)^{-1} \mathcal {Y}_{m-1,s}(\mathbf {W},\mathbf {B}) \nabla \sigma _k(X_s^{x,a}) \mathop {}\!\mathrm {d}\mathbf {B}_s^k\right),\) 
Then, for any \(m\ge 1\)  and \(h\in \mathcal {H}([a,b])\) ,
\(\tilde{D}^m_h\mathcal {X}_{m,t}(\mathbf {W},\cdot ) = m! D^m_h X_t^{x,a}\quad \mbox{and}\quad \tilde{D}^m_h\mathcal {Y}_{m,t}(\mathbf {W},\cdot ) = m! D^m_h (\mathbf {J}_t^{x,a})^{-1},\) 
where \(\tilde{D}\)  is the Malliavin derivative with respect to \(B\)  and the left-hand side does not depend on \(\mathbf {B}\) , and we have the estimates, for any \(r\ge 2\) ,
\(\mathbb {E}\left[ { \sup _{x}}\left\Vert  D^m X_t^{x,a}\right\Vert _{\mathcal {H}^{\otimes m}([a,b])}^r \right] \lesssim _{m,r} \mathbb {E}\left[{ \sup _{x}}\left| \mathcal {X}_{m,t}(\mathbf {W},\mathbf {B})\right|^r \right]\\\mbox{and}\quad \mathbb {E}\left[{ \sup _{x}}\left\Vert  D^m (\mathbf {J}_t^{x,a})^{-1}\right\Vert _{\mathcal {H}^{\otimes m}([a,b])}^r \right] \lesssim _{m,r} \mathbb {E}\left[{ \sup _{x}}\left| \mathcal {Y}_{m,t}(\mathbf {W},\mathbf {B})\right|^r \right].\) 

We only give the main arguments of the proofs (see [15]}). The fact that the Malliavin derivative with respect to \(B\)  of \(\mathcal {X}_m\)  (resp. \(\mathcal {Y}\) ) is related to the Malliavin derivative of \(X\)  (resp. \(\mathbf {J}^{-1}\) ) is a straightforward consequence of their expressions. Concerning the inequality, we remark that
\(\left\Vert  D^m X_t^{x,a}\right\Vert _{\mathcal {H}^{\otimes m}([a,b])} = \frac{1}{m!} \left\Vert \tilde{D}^m\mathcal {X}_{m,t}(\mathbf {W},\cdot )\right\Vert _{\mathcal {H}^{\otimes m}([a,b])} = \frac{1}{m!}\tilde{\mathbb {E}}\left[ \left\Vert \tilde{D}^m\mathcal {X}_{m,t}(\mathbf {W},\cdot )\right\Vert _{\mathcal {H}^{\otimes m}([a,b])}^2 \right]^{\frac{1}{2}}\) 
where \(\tilde{\mathbb {E}}\)  is the expectation with respect to \(B\) , as well as
\(\left\Vert  D^m (\mathbf {J}_t^{x,a})^{-1}\right\Vert _{\mathcal {H}^{\otimes m}([a,b])} = \frac{1}{m!} \left\Vert \tilde{D}^m\mathcal {Y}_{m,t}(\mathbf {W},\cdot )\right\Vert _{\mathcal {H}^{\otimes m}([a,b])} = \frac{1}{m!}\tilde{\mathbb {E}}\left[ \left\Vert \tilde{D}^m\mathcal {Y}_{m,t}(\mathbf {W},\cdot )\right\Vert _{\mathcal {H}^{\otimes m}([a,b])}^2 \right]^{\frac{1}{2}}.\) 
Since \(\mathcal {X}_m\)  (resp. \(\mathcal {Y}_m\) ) belongs to the \(m\) -th order inhomogeneous Wiener chaos generated by \(B\) , we know that all the \(\mathbb {D}^{2,m}\) -norms are equivalent for the right-hand-side term. Hence, we obtain
\(\mathbb {E}\left[ { \sup _{x}} \left\Vert  D^m X_t^{x,a}\right\Vert _{\mathcal {H}^{\otimes m}([a,b])}^r \right]&\lesssim _{m,r} \mathbb {E}\left[{ \sup _{x}}\tilde{\mathbb {E}}\left[ \left\Vert \tilde{D}^m\mathcal {X}_{m,t}(\mathbf {W},\cdot )\right\Vert _{\mathcal {H}^{\otimes m}([a,b])}^2 \right]^{r/2} \right]\\ &\lesssim _{m,r} \mathbb {E}\left[\tilde{\mathbb {E}}\left[ { \sup _{x}}\left|\mathcal {X}_{m,t}(\mathbf {W},\mathbf {B})\right|^2 \right]^{r/2} \right],\) 

and, similarly,
\(\mathbb {E}\left[ { \sup _{x}}\left\Vert  D^m (\mathbf {J}_t^{x,a})^{-1}\right\Vert _{\mathcal {H}^{\otimes m}([a,b])}^r \right]\lesssim _{m,r} \mathbb {E}\left[\tilde{\mathbb {E}}\left[ { \sup _{x}}\left|\mathcal {Y}_{m,t}(\mathbf {W},\mathbf {B})\right|^2 \right]^{r/2} \right],\) 
for which gives the desired results.
In order to push further estimates (REF ) and (), we need to estimate the rough integrals from (REF ), (REF ), (REF ) and (REF ). This is done by considering, for any \(m\ge 1\) , the function \((X,\mathbf {J},\mathbf {J}^{-1}, \mathcal {X}_1,\mathcal {X}_2,\ldots ,\mathcal {X}_m, \mathcal {Y}_1,\mathcal {Y}_2,\ldots ,\mathcal {Y}_m)\) 
as a solution of a (system of) RDE given by (REF )-(REF )-(REF )-(REF )-(REF )-(REF )-(REF ) driven by \(\mathbf {Z}\)  which is the lifted Gaussian process \(Z = (W,B)\)  (which satisfies Assumption REF ) in the geometric \(p\) -rough paths. It turns out that, for any \(q>2\)  large enough, we have estimate [18]}
\(\mathbb {E}\left[\Vert \mathbf {Z}\Vert _{p\textrm {-var};[a,t]}^q\right] \lesssim _q \kappa _{a,t}.\) 
We remark that we can not immediately apply the estimate (REF ) since the vector-fields in (REF )-(REF )-(REF )-(REF )-(REF )-(REF ) are not bounded with respect to \((\mathbf {J},\mathbf {J}^{-1}, \mathcal {X}_1,\mathcal {X}_2,\ldots ,\mathcal {X}_m,\)  \( \mathcal {Y}_1,\mathcal {Y}_2,\ldots ,\mathcal {Y}_m)\)  but have (as well as their derivatives) polynomial growth.
To proceed, as in [8]}, we use an induction argument. We first consider \(V = (X,Y)\)  where, for any \(1\le k\le d\) ,
\((Y_t)^k = \int _a^t \nabla \sigma _k(X_s^{x,a})d\mathbf {W}_s^k.\) 
Then, \(V\)  is the solution of (REF )-(REF ) driven by \(\mathbf {Z}\)  and, thus, is a geometric \(p\) -rough path (denoted \(\mathbf {V}\) ). Since the vector-fields are smooth and bounded, we have, by Theorem REF ,
\(\Vert \mathbf {V}\Vert _{p\textrm {-var};[a,t]} \lesssim _{p,\gamma ,\sigma }\Vert \mathbf {Z}\Vert _{p\textrm {-var};[a,t]}.\) 
Remark 5.21 Note here that when \(x\)  is smooth, the definition of \(S_{\lfloor p \rfloor }(x)\)  only involved increments of \(x\) . Remark also that when \(W^\varepsilon \)  is a sequence of smooth paths such that \(S_{\lfloor p \rfloor }(W^\varepsilon )\)  converge to \(\mathbf {W}\)  in the rough path topology, then by Inequality (REF ),
\(\sup _{x } \Vert  S_{\lfloor p \rfloor }(X^{\varepsilon ,s,x})\Vert _{p\textrm {-var};[s,t]}^p \lesssim \left(\Vert \sigma \Vert _ {\mathcal {C}^{\gamma -1}_b}\Vert S_{\lfloor p \rfloor }(W^\varepsilon )\Vert _{p\textrm {-var};[s,t]} \vee \Vert \sigma \Vert _ {\mathcal {C}^{\gamma -1}_b}^p\Vert S_{\lfloor p \rfloor }(W^\varepsilon )\Vert _{p\textrm {-var};[s,t]}^p\right)\) 
and one has, by letting \(\varepsilon \rightarrow 0\) ,
\(\sup _{x}\Vert  \mathbf {X}^{s,x}\Vert _{p\textrm {-var};[s,t]}^p\lesssim \left(\Vert \sigma \Vert _ {\mathcal {C}^{\gamma -1}_b}\Vert \mathbf {W}\Vert _{p\textrm {-var};[s,t]}\vee \Vert \sigma \Vert _ {\mathcal {C}^{\gamma -1}_b}^p\Vert \mathbf {W}\Vert _{p\textrm {-var};[s,t]}^p\right)\) 

Now, let \(V_1 = (X,\mathbf {J},\mathbf {J}^{-1})\)  be a solution of the RDE
(REF )-(REF )-(REF ) driven by \(\mathbf {V}\) . This RDE can be solved and is a geometric \(p\) -rough path (denoted \(\mathbf {V}_1\) ) that satisfies [14]}, [21]}
\(\Vert \mathbf {V}_1\Vert _{p\textrm {-var};[s,t]} \le \kappa _1 \Vert \mathbf {V}\Vert _{p\textrm {-var};[s,t]}e^{\kappa _2 N_{\alpha ,[s,t],p}(\mathbf {V})},\) 
where \(\kappa _1,\kappa _2>0\)  depend on \(\sigma \)  and \(N_{\alpha ,[s,t],p}(\mathbf {V})\)  as finite moments of any order (see [22]}, [21]} for details). Now, we can see that the integrand in (REF ) as a polynomial growth (as well as its derivatives) with respect to \(\mathbf {V}_1\) . Thus, we have the following estimate from [8]} (see also [21]} Subsection 4.3)
\(\left| \mathcal {X}_{1,t}(\mathbf {W},\mathbf {B})\right|\wedge \left\Vert  \mathcal {X}_{1}(\mathbf {W},\mathbf {B})\right\Vert _{p\textrm {-var};[a,t]}\lesssim \left( 1+\Vert \mathbf {V}_1\Vert _{p\textrm {-var};[a,t]}\right)^r\Vert \mathbf {V}_1\Vert _{p\textrm {-var};[a,t]}\) 
for some \(r>0\) . This yields, in the end,
\(\left| \mathcal {X}_{1,t}(\mathbf {W},\mathbf {B})\right|\lesssim \left( 1+\Vert \mathbf {Z}\Vert _{p\textrm {-var};[a,t]}e^{\kappa _2 N_{\alpha ,[a,t],p}(\mathbf {V})}\right)^r\Vert \mathbf {Z}\Vert _{p\textrm {-var};[a,t]}e^{\kappa _2 N_{\alpha ,[a,t],p}(\mathbf {V})},\) 
which leads to the estimate
\(\mathbb {E}\left[{ \sup _{x}}\left| \mathcal {X}_{1,t}(\mathbf {W},\mathbf {B})\right|^q\right] \lesssim \kappa _{a,t}.\) 
Finally, if we assume that we have, for any \(1\le \ell \le m-1\) ,
\(\left\Vert  \mathcal {X}_{\ell }(\mathbf {W},\mathbf {B})\right\Vert _{p\textrm {-var};[a,t]} \le \left( 1+\Vert \mathbf {Z}\Vert _{p\textrm {-var};[a,t]}e^{\kappa _2 N_{\alpha ,[a,t],p}(\mathbf {V})}\right)^r\Vert \mathbf {Z}\Vert _{p\textrm {-var};[a,t]}e^{\kappa _2 N_{\alpha ,[a,t],p}(\mathbf {V})},\) 
for some \(r>0\) , then one can consider the function \(V_{m-1} = (X,\mathbf {J},\mathbf {J}^{-1},\mathcal {X}_1,\ldots ,\mathcal {X}_{m-1})\)  as a geometric \(p\) -rough path driven by \(V\)  denoted by \(\mathbf {V}_{m-1}\)  and, since the integrand in (REF ) as a polynomial growth (as well as its derivatives), we deduce the estimate
\(\left|\mathcal {X}_{m,t}(\mathbf {W},\mathbf {B})\right|\wedge \left\Vert  \mathcal {X}_{m}(\mathbf {W},\mathbf {B})\right\Vert _{p\textrm {-var};[a,t]}\lesssim \left( 1+\Vert \mathbf {V}_{m-1}\Vert _{p\textrm {-var};[a,t]}\right)^r\Vert \mathbf {V}_{m-1}\Vert _{p\textrm {-var};[a,t]}.\) 
In particular, following the same arguments as for \(\mathcal {X}_1\) , we obtain that
\(\mathbb {E}\left[{ \sup _{x}}\left| \mathcal {X}_{m,t}(\mathbf {W},\mathbf {B})\right|^q\right] \lesssim \kappa _{a,t}.\) 
We proceed then to obtain estimates on \((\mathcal {Y}_{\ell })_{1\le \ell \le m}\)  by considering iteratively \(V_{\ell +m} = (X,\mathbf {J},\mathbf {J}^{-1},\)  \(\mathcal {X}_1,\ldots ,\mathcal {X}_{m},\mathcal {Y}_{1},\ldots ,\mathcal {Y}_{\ell })\)  taken as a geometric \(p\) -rough path driven by \(V\)  and denoted \(\mathbf {V}_{\ell +m}\) . By considering (REF )-(REF ), we deduce that, for any \(0\le \ell \le m-1\) ,
\(\left|\mathcal {Y}_{\ell +1,t}(\mathbf {W},\mathbf {B})\right|\wedge \left\Vert  \mathcal {Y}_{\ell +1}(\mathbf {W},\mathbf {B})\right\Vert _{p\textrm {-var};[a,t]}\lesssim \left( 1+\Vert \mathbf {V}_{\ell +m}\Vert _{p\textrm {-var};[a,t]}\right)^r\Vert \mathbf {V}_{\ell +m}\Vert _{p\textrm {-var};[a,t]},\) 
which leads to the estimate
\(\mathbb {E}\left[{ \sup _{x}}\left| \mathcal {Y}_{m,t}(\mathbf {W},\mathbf {B})\right|^q\right] \lesssim \kappa _{a,t}.\) 
For the following result, we need a notation. Let \(F:x\mapsto F(x) \in \mathbb {D}^{m,p}\)  Let us define
\(\Vert F\Vert _{\star ,\mathbb {D}^{m,p}}= \left( \mathbb {E}[\sup _{x\in \mathbb {R}^d}|F(x)|^p] + \sum _{k = 1}^m \mathbb {E}[\sup _{x\in \mathbb {R}^d}\Vert D^k F(x)\Vert _{\mathcal {H}^{\otimes k}}^p]\right)^{1/p}.\) 
What we just have proved is that
Proposition 5.22 
We have the estimates, for any \(m\ge 1\)  and \(p\ge 2\) ,
\(\Vert DX_t^{x,a}\Vert _{\star ,\mathbb {D}^{m,p}(\mathcal {H}([a,b]))}\lesssim \kappa _{a,t}\quad \mbox{and}\quad \Vert (\mathbf {J}_t^{x,a})^{-1}\Vert _{\star ,\mathbb {D}^{m,p}(\mathcal {H}([a,b]))}\lesssim \kappa _{a,t}.\) 

Remark 5.23 
Let us remark that if if we denote by \(\mathbf {J}^0 = \mathbf {J}\)  and \(\mathbf {J}^m = D_m \mathbf {J}^{k-1}\) , where \(D_x\)  stands for the derivative with respect to \(x\) . We have some constants such that following equation :
\( \mathbf {J}^{m}_{t} = \sum _{\begin{array}{c}\ell \in \lbrace  2,\cdots ,m\rbrace \\ \alpha \in \lbrace 1,\cdots ,m-1\rbrace ^{\ell } \\ |\alpha | = m\end{array}} c_{\ell ,\alpha }\sum _{k=1}^d D^{\ell }\sigma _k(X^{a,x}_r) \mathbf {J}^{\alpha _1}_r \cdots \mathbf {J}^{\alpha _\ell }_r \mathop {}\!\mathrm {d}\mathbf {W}_r + \sum _{k=1}^{d} \int _a^t \nabla \sigma _k(X^{a,x}_r) \mathbf {J}^{m}_{r} \mathop {}\!\mathrm {d}\mathbf {W}_r.\) 
By using exactly the same strategy as before, one can also prove that
\(\Vert \mathbf {J}^m_t\Vert _{\star ,\mathbb {D}^{k,q}(\mathcal {H}([a,b]))} \lesssim \kappa _{a,t}\) 
Indeed one has just to reuse Equation (REF ) in this situation.

We now turn to the covariance matrix \(\gamma _{X,[a,b]}\)  which is expressed, following Definition REF , as
\(\gamma _{X_t^{x,a},[a,b]} = \left( \langle DX^{x,a,i}_t,DX^{x,a,j}_t \rangle _{\mathcal {H}([a,b])}\right)_{(i,j)\in \lbrace 1,\ldots ,d\rbrace ^2}.\) 
We remark that \(\gamma _{X_t,[a,b]}\)  is a symmetric definite positive matrix. Furthermore, thanks to (REF ), (REF ) and (REF ), we deduce the expression
\(\gamma _{X_t^{x,a},[a,b]} &= \langle DX^{x,a}_t,(DX^{x,a}_t)^* \rangle _{\mathcal {H}([a,b])} = \langle DX^{x,a}_t,(DX^{x,a}_t)^* \rangle _{\mathcal {H}([a,t])}\\ &= \int _a^t \int _a^t \mathbf {J}_{t}^{x,s_1}\sigma (X_{s_1}^{x,a})\sigma (X_{s_2}^{x,a})^* \left(\mathbf {J}_{t}^{x,s_2}\right)^* \mathop {}\!\mathrm {d}R_W(s_1,s_2).\) 
Proposition 5.24 
Under the uniform ellipticity condition
\(\Vert \sigma (x)z\Vert ^2\ge \varsigma \Vert z\Vert ^2,\quad \forall z,x\in \mathbb {R}^d,\) 
for some constant \(\varsigma >0\) , we have the estimate, for any \(m\ge 1\)  and \(p\ge 2\) ,
\(\left\Vert \gamma _{X_t^{x,a},[a,b]}^{-1} \right\Vert _{\star ,\mathbb {D}^{m,p}(\mathcal {H}([a,b]))}\lesssim \frac{\max (1,\varrho _{a,t})^m}{\sigma _{a,t}^2},\) 
where \(\varrho _{a,t} : = \kappa _{a,t}/\sigma _{a,t}\) .

We follow the arguments from [8]}. We start with the estimate in the case \(m=0\) . Since \(\gamma _{X_t^{x,a},[a,b]}\)  is symmetric definite positive and all the matrix norms are equivalent, an upper bound on \(\Vert \gamma _{X_t^{x,a},[a,b]}^{-1} \Vert \)  can be deduced by estimating its lowest eigenvalue. In order to do so, we will identify a positive random variable \(\varrho _{a,t}\)  admitting negative moments of any order such that, for any \(z\in \mathbb {R}^d\backslash \lbrace 0\rbrace \) ,
\(z^*\, \gamma _{X_t^{x,a},[a,b]}\,z \ge \varrho _{a,t} \Vert z\Vert ^2.\) 
Thanks to (REF ) and by denoting \(\mathsf {f}_s^{z,x,a,t} = \sigma (X_{s}^{x,a})^* \left(\mathbf {J}_{t}^{x,s}\right)^* z \) , we obtain
\(z^*\, \gamma _{X_t^{x,a},[a,b]}\,z &= \int _a^t \int _a^t z^* \mathbf {J}_{t}^{x,s_1}\sigma (X_{s_1}^{x,a})\sigma (X_{s_2}^{x,a})^* \left(\mathbf {J}_{t}^{x,s_2}\right)^*z \mathop {}\!\mathrm {d}R_W(s_1,s_2)\\ &=\int _a^t \int _a^t\left(\mathsf {f}_{s_1}^{z,x,a,t}\right)^* \mathsf {f}_{s_2}^{z,x,a,t} \mathop {}\!\mathrm {d}R_W(s_1,s_2) = \Vert \mathsf {f}^{z,x,a,t}\Vert ^2_{\mathcal {H}([a,t])}.\) 

By Corollary REF , we furthermore deduce that
\(\Vert \mathsf {f}^{z,x,a,t}\Vert ^2_{\mathcal {H}([a,t])} \ge \frac{\sigma _{a,t}^2 \Vert \mathsf {f}^{z,x,a,t}\Vert _{\infty ;[a,t]}^2}{4}\min \left(1,\frac{4(c_W/2)^{(2\gamma +\alpha )/2\gamma }\Vert \mathsf {f}^{z,x,a,t}\Vert _{\infty ;[a,t]}^{\alpha /\gamma }}{\sigma _{a,t}^2|\mathsf {f}^{z,x,a,t}|_{\mathcal {C}^{\gamma }([a,t])}^{\alpha /\gamma }} \right).\) 
Also, thanks to the uniform ellipticity condition, we obtain
\(\Vert \mathsf {f}^{z,x,a,t}_s\Vert ^2 \ge \varsigma \Vert \left(\mathbf {J}_{t}^{x,s}\right)^{-1}z\Vert ^2\ge \varsigma \Vert \mathbf {J}_{t}^{x,s}\Vert ^{-2} \Vert z\Vert ^2,\) 
which, since \(\sup _{s\in [a,t]} \Vert \mathbf {J}_{t}^{x,s}\Vert ^{-1}\ge \Vert \mathbf {J}_{t}^{x,t}\Vert ^{-1} = \Vert \mathrm {Id} \Vert ^{-1} = 1\) , writes as
\(\Vert \mathsf {f}^{z,x,a,t}\Vert _{\infty ;[a,t]} \ge \sqrt{\varsigma } \Vert z\Vert .\) 
Moreover, we have
\(|\mathsf {f}^{z,x,a,t}|_{\mathcal {C}^{\gamma }([a,t])} \le | \sigma (X_{\cdot }^{x,a})^* \left(\mathbf {J}_{t}^{x,\cdot }\right)^*|_{\mathcal {C}^{\gamma }([a,t])} \Vert z\Vert  = | \mathbf {J}_{t}^{x,\cdot }\sigma (X_{\cdot }^{x,a})|_{\mathcal {C}^{\gamma }([a,t])} \Vert z\Vert .\) 
This leads to the estimate
\(z^*\, \gamma _{X_t^{x,a},[a,b]}\,z \ge \frac{\sigma _{a,t}^2 \varsigma }{4}\min \left(1,\frac{4\varsigma ^{\alpha /\gamma } (c_W/2)^{(2\gamma +\alpha )/2\gamma }}{\sigma _{a,t}^2|\mathbf {J}_{t}^{x,\cdot }\sigma (X_{\cdot }^{x,a})|_{\mathcal {C}^{\gamma }([a,t])}^{\alpha /\gamma }} \right) \Vert z\Vert ^2,\) 
and we identify
\(\varrho _{a,t} = \frac{\sigma _{a,t}^2 \varsigma }{4}\inf _{x\in \mathbb {R}^d}\left(\min \left(1,\frac{4\varsigma ^{\alpha /\gamma } (c_W/2)^{(2\gamma +\alpha )/2\gamma }}{\sigma _{a,t}^2|\mathbf {J}_{t}^{x,\cdot }\sigma (X_{\cdot }^{x,a})|_{\mathcal {C}^{\gamma }([a,t])}^{\alpha /\gamma }} \right)\right).\) 
We now have to prove that \(\mathbb {E}[ { \sup _{x\in \mathbb {R}^d}}\varrho _{a,t}^{-p}]<+\infty \) . We can see that
\(\varrho _{a,t}^{-1} \le \frac{4}{\sigma _{a,t}^2 \varsigma }\sup _{x\in \mathbb {R}^d}\left(\max \left(1,\frac{\sigma _{a,t}^2|\mathbf {J}_{t}^{x,\cdot }\sigma (X_{\cdot }^{x,a})|_{\mathcal {C}^{\gamma }([a,t])}^{\alpha /\gamma }}{4\varsigma ^{\alpha /\gamma } (c_W/2)^{(2\gamma +\alpha )/2\gamma }} \right)\right),\) 
and, thus, we essentially have to see that \(\sup _{x\in \mathbb {R}^d}|\mathbf {J}_{t}^{x,\cdot }\sigma (X_{\cdot }^{x,a})|_{\mathcal {C}^{\gamma }([a,t])}\)  admits moments of any order. We have
\(\sup _{x\in \mathbb {R}^d}|\mathbf {J}_{t}^{x,\cdot }\sigma (X_{\cdot }^{x,a})|_{\mathcal {C}^{\gamma }([a,t])} \le \Vert \sigma \Vert _{\mathcal {C}^0_b}\sup _{x\in \mathbb {R}^d}|\mathbf {J}_{t}^{x,\cdot }|_{\mathcal {C}^{\gamma }([a,t])} + \sup _{s\in [a,t],\,x\in \mathbb {R}^d}|\mathbf {J}_{t}^{x,s}| \sup _{x\in \mathbb {R}^d}|\sigma (X_{\cdot }^{x,a})|_{\mathcal {C}^{\gamma }([a,t])}.\) 
From here, it is rather direct to prove that, for any \(p\in [2,+\infty )\) ,
\(\mathbb {E}\left[ \left(\sup _{x\in \mathbb {R}^d}|\sigma (X_{\cdot }^{x,a})|_{\mathcal {C}^{\gamma }([a,t])}\right)^p\right] \le \Vert \sigma \Vert _{\mathcal {C}^1_b}\mathbb {E}\left[\left( \sup _{x\in \mathbb {R}^d}|X_{\cdot }^{x,a}|_{\mathcal {C}^{\gamma }([a,t])}\right)^p\right]<+\infty ,\) 
thanks to (REF ) and Proposition REF . We use Equation (REF ) to handle the Jacobian. In the end, we obtain the bound
\(\mathbb {E}\left[ \varrho _{a,t}^{-p}\right] \lesssim \sigma _{a,t}^{-2p},\) 
which provides the desired result in the case \(m = 0\) . In the case \(m\ge 1\) , we use Lemma REF  to deduce that, for any \((i,j)\in \lbrace 1,\ldots ,d\rbrace ^2\) ,
\(D(\gamma _{X_t^{x,a},[a,b]}^{-1})_{i,j} = -\sum _{\ell _1,\ell _2 = 1}^d (\gamma ^{-1}_{X_t^{x,a},[a,b]})_{i,\ell _1}\,D(\gamma _{X_t^{x,a},[a,b]})_{\ell _1,\ell _2}\,(\gamma ^{-1}_{X_t^{x,a},[a,b]})_{\ell _2,j},\) 
which yields, thanks to Leibniz's rule, for any \(m\ge 1\) ,
\(D^m(\gamma ^{-1}_{X_t^{x,a},[a,b]})_{i,j}=-\sum _{\ell _1,\ell _2 = 1}^d\sum _{\begin{array}{c}k_1,k_2,k_3\in \mathbb {N}\\k_1+k_2+k_3 = m-1\end{array}}\begin{pmatrix}m-1\\ k_1,k_2,k_3\end{pmatrix} \\ D^{k_1}(\gamma ^{-1}_{X_t^{x,a},[a,b]})_{i,\ell _1}\,D^{k_2+1}(\gamma _{X_t^{x,a},[a,b]})_{\ell _1,\ell _2}\,D^{k_3}(\gamma ^{-1}_{X_t^{x,a},[a,b]})_{\ell _2,j}.\) 
From here, we can see that, thanks to (REF ),
\(D^{k_2+1}(\gamma _{X_t^{x,a},[a,b]})_{\ell _1,\ell _2} = \sum _{k = 0}^{k_2+1}\begin{pmatrix}k_2+1\\ k\end{pmatrix}\langle D^{k+1}X^{x,a}_t,(D^{k_2+1-k}X^{x,a}_t)^* \rangle _{\mathcal {H}([a,b])},\) 
which provides the estimate, thanks to Proposition REF  and Hölder's inequality, for any \(p\in [2,+\infty )\) ,
\(\mathbb {E}\left[{ \sup _{x\in \mathbb {R}^d}}\Vert D^{k_2+1}(\gamma _{X_t^{x,a},[a,b]})_{\ell _1,\ell _2}\Vert _{\mathcal {H}^{\otimes (k_2+1)}([a,b])}^p \right]\\ \le \sum _{k = 0}^{k_2+1} \begin{pmatrix}k_2+1\\ k\end{pmatrix} \mathbb {E}\left[{ \sup _{x\in \mathbb {R}^d}}\Vert D^{k+1}X^{x,a}_t\Vert _{\mathcal {H}^{\otimes (k+1)}([a,b])}^{p_1}\right]^{1/p_1} \\ \times \mathbb {E}\left[{ \sup _{x\in \mathbb {R}^d}}\Vert D^{k_2+1-k}X^{x,a}_t\Vert _{\mathcal {H}^{\otimes (k_2+1-k)}([a,b])}^{p_2}\right]^{1/p_2}\lesssim \kappa _{a,t}^2,\) 
where \(1/p = 1/p_1+1/p_2\) .
Proceeding by induction, we assume that, for any \(p\in [2,+\infty )\) ,
\(\mathbb {E}\left[{ \sup _{x\in \mathbb {R}^d}}\Vert D^{m-1}(\gamma ^{-1}_{X_t^{x,a},[a,b]})\Vert _{\mathcal {H}^{\otimes (m-1)}([a,b])}^p\right]^{1/p} \lesssim \frac{\max (1,\varrho _{a,t})^{m-1}}{\sigma _{a,t}^2}.\) 
From (REF ), (REF ) and Hölder's inequality, we deduce that, for \(1/p = 1/p_1+1/p_2+1/p_3\) ,
\(&\mathbb {E}\left[{ \sup _{x\in \mathbb {R}^d}}\Vert D^{m}(\gamma ^{-1}_{X_t^{x,a},[a,b]})\Vert _{\mathcal {H}^{\otimes m}([a,b])}^p\right]^{1/p}\\ &\lesssim \sum _{\begin{array}{c}k_1,k_2,k_3\in \mathbb {N}\\k_1+k_2+k_3 = m-1\end{array}}\begin{pmatrix}m-1\\ k_1,k_2,k_3\end{pmatrix} \mathbb {E}\left[{ \sup _{x\in \mathbb {R}^d}}\Vert D^{k_1}(\gamma ^{-1}_{X_t^{x,a},[a,b]})\Vert _{\mathcal {H}^{\otimes k_1}([a,b])}^{p_1}\right]^{1/p_1}\\ & \hspace{20.0pt}\times \mathbb {E}\left[{ \sup _{x\in \mathbb {R}^d}}\Vert D^{k_2+1}\gamma _{X_t^{x,a},[a,b]}\Vert _{\mathcal {H}^{\otimes (k_2+1)}([a,b])}^{p_2}\right]^{1/p_2}\mathbb {E}\left[{ \sup _{x\in \mathbb {R}^d}}\Vert D^{k_3}(\gamma ^{-1}_{X_t^{x,a},[a,b]})\Vert _{\mathcal {H}^{\otimes k_3}([a,b])}^{p_3}\right]^{1/p_3},\\ &\lesssim \sum _{\begin{array}{c}k_1,k_2,k_3\in \mathbb {N}\\k_1+k_2+k_3 = m-1\end{array}}\begin{pmatrix}m-1\\ k_1,k_2,k_3\end{pmatrix}\frac{\max (1,\varrho _{a,t})^{k_1+k_3}}{\sigma _{a,t}^2}\lesssim \frac{\max (1,\varrho _{a,t})^{m}}{\sigma _{a,t}^2},\) 

which concludes our proof.

Proof of the main theorem
Theorem 6.1 
Let \(W\)  be a Gaussian process which satisfies Assumption REF , REF  and REF . Let \(2<p<4\)  such that \(W\)  can be lift into a geometric \(p\)  rough path (from assumption (REF )). Furthermore let \(\alpha \)  its index of non-determinism from Assumption REF .
Let \(\sigma \in C^\infty _{b}(\mathbb {R}^d;\mathbb {R}^{d\times d})\)  be such there exists \(\varsigma >0\)  that for all \(x,z\in \mathbb {R}^d\)
\( |\sigma (x) z|^2 \ge \varsigma |z|^2.\) 
Let \(b\in { \mathcal {C}^{\kappa }}(\mathbb {R}^d\)  with \(\kappa + \frac{1}{\alpha }>\frac{1}{2}\) . Then, almost surely, there exists a solution flow to the equation
\(\mathop {}\!\mathrm {d}X_t = b(X_t) \mathop {}\!\mathrm {d}t + \sigma (X_t) \mathop {}\!\mathrm {d}\mathbf {W}_t, \quad X_0 = x,\quad t\in [0,T].\) 
Furthermore, this flow is locally Lipschitz continuous.

As seen in Section  and especially in Theorem REF , one only has to check that there exists \(\nu >\frac{1}{2}\) , \(\kappa >2-\nu \)  and a weight \(w_0\)  such that almost surely
\(T^{D\varphi ,\varphi } b \in \mathcal {C}^{\nu }_T \mathcal {C}^{\kappa }_{w_0}.\) 
Furthermore, thanks to Section  and especially Corollary REF , it is enough to check that for all \(i,j\in \lbrace 1,\cdots ,d\rbrace \) , we have the following couples
\((\mathbf {J}^{-1},X),(\partial _i \mathbf {J}^{-1} ,X), (\partial _i X \mathbf {J}^{-1},X), (\partial _j \partial _i X \mathbf {J}^{-1} ,X), \partial _j \partial _i \mathbf {J}^{-1}, \partial _j \partial _i X \mathbf {J}^{-1}, \partial _j X \partial _i X \mathbf {J}^{-1},X) \) 
which satisify Assumption REF  for some \(H\in (0,1)\) .
Remark that thanks to the flow properties, for \(s\in [0,T)\)  one can always consider that for \(t\in [s,T]\) ,
\(\phi (\cdot ,x) \in \lbrace  (\mathbf {J}^{x,s}_{\cdot })^{-1}, \partial _i (\mathbf {J}^{x,s}_{\cdot })^{-1}, \partial _i X^{s,x}_{\cdot } (\mathbf {J}^{x,s}_{\cdot })^{-1},\\\partial _i\partial _j(\mathbf {J}^{x,s}_{\cdot })^{-1} \partial _i X^{x,s}_{\cdot } \partial _j (\mathbf {J}^{x,s}_{\cdot })^{-1}, \partial _i \partial _j X^{x,s}_{\cdot }(\mathbf {J}^{x,s}_{\cdot })^{-1}, \partial _i X^{s,x}_{\cdot } \partial _j X^{s,x}_{\cdot } (\mathbf {J}^{x,s}_{\cdot })^{-1}\rbrace \) 
and
\(\varphi (\cdot ,x) = X^{s,x}_{\cdot }.\) 
Thanks to Proposition REF , for all \(t \in [0,T]\) , we take \(F = X_t^{s,x}\)  and \(G=\phi (t,x)\)  as previously. Remark that \(F, G \in \mathcal {F}_t\) . We have for all \(\beta \in \mathbb {N}^d\)  and all \(f\in \mathcal {S}\) ,
\(\mathbb {E}[\partial ^{\beta }f(F) G |\mathcal {F}_s] =\mathbb {E}[f(F) H_{\beta ,[s,t]}(F,G) |\mathcal {F}_{[0,s]}].\) 
and for all \(q\ge 0\) ,
\(|\mathbb {E}[\partial ^{\beta }f(F) G |\mathcal {F}_s]|=&|\mathbb {E}[f(F) H_{\beta ,[s,t]}(F,G) |\mathcal {F}_{[0,s]}]|\le \mathbb {E}[f(S)^2|\mathcal {F}_s]^{\frac{1}{2}}\mathbb {E}[|H_{\beta ,[s,t]}(F,G)|^2|\mathcal {F}_s]^{\frac{1}{2}} \\\le & \Vert f\Vert _\infty { \sup _{x\in \mathbb {R}^d}} \mathbb {E}[|H_{\beta ,[s,t]}(F,G)|^2|\mathcal {F}_s]^{\frac{1}{2}}\) 
Note that for \(q\ge 2\) , thanks to Lemma REF ,
\(\Vert { \sup _{x\in \mathbb {R}^d}}\mathbb {E}[H_{\beta ,[s,t]}(F,G)^2|\mathcal {F}_s]^\frac{1}{2}\Vert _{L^q(\Omega )}\le &\Vert { \sup _{x\in \mathbb {R}^d}}|H_{\beta ,[s,t]}(F,G)| \Vert _{L^{\frac{q}{2}}(\Omega )}^{\frac{1}{2}}\\\le & \Vert G\Vert _{\star ,\mathbb {D}^{|\beta |,r}}^{\frac{1}{2}} \left(\prod _{\ell = 1}^{|\beta |} \left\Vert \sum _{j =1}^d (\gamma _{X^{s,x}_t,[s,t]}^{-1})_{\beta _\ell ,j}D(X^{s,x}_t)^j \right\Vert _{\star ,\mathbb {D}^{|\beta |-\ell +1,r_\ell }}\right)^{\frac{1}{2}}\) 

with \(\frac{2}{q}=\frac{1}{r} + \sum \frac{1}{r_i}\) .
Furthermore, thanks to Proposition REF  and Remark REF , we know that for every \(\beta \in \mathbb {N}^d\)  and every \(r\ge 1\)
\(\Vert |G|\Vert _{\star ,\mathbb {D}^{|\beta |,r}} \lesssim \kappa _{s,t} = \Vert R_W\Vert _{(1,\rho )\textrm {-var};[a,b]^2}^{1/2} \lesssim _T 1.\) 
Furthermore, Thanks to Propositions REF  REF  and Hölder inequaltiy, , we know that for all \(\ell \le |\beta |\)  and all \(r\ge 2\) ,
\(\Vert \gamma ^{-1}_{X^{x,s}_t,[a,b]} (D(X^{s,x}_t)^j \Vert _{\star ,\mathbb {D}^{|\beta |-\ell + 1,r}} \lesssim \frac{1}{\sigma _{a,b}^{2}},\) 
and finally
\( \Vert {  \sup _{x\in \mathbb {R}^d} }\mathbb {E}[H_{\beta ,[s,t]}(F,G)^2|\mathcal {F}_s]^\frac{1}{2}\Vert _{L^q(\Omega )} \lesssim \sigma _{s,t}^{-|\beta |}.\) 
Furthermore, note that since \(W\)  is centered and has the local non-determinism property, we know that
\(\sigma _{s,t}^2 = \mathbb {E}[(W_t-W_s)^2] = \mathbb {E}\left[\mathbb {E}[(W_t-W_s)^2|\mathcal {F}_{[0,s]}\vee \mathcal {F}_{{t,T}}]\right] \gtrsim |t-s|^{\alpha },\) 
where \(\alpha \)  is the index of non-determinism. Altogether, it gives
\(|\mathbb {E}[\partial ^{\beta } f(F) G | \mathcal {F}_s]| \lesssim \Vert f\Vert _{\infty }|t-s|^{-\frac{\alpha }{2} |\beta |} G_{\infty ,s},\) 
where
\(G_{\infty ,s} = (t-s)^{\frac{\alpha }{2} |\beta |} \sup _{x\in \mathbb {R}^d} \mathbb {E}[H_{\beta ,[s,t]}(F,G)^2|\mathcal {F}_s]^{\frac{1}{2}}.\) 
enjoys finite \(L^q(\Omega )\)  moments which are bounded uniformly in \((s,t)\in \Delta ^2_T\) .
Hence, using Corollary REF , for all \(\kappa \in \mathbb {R}\)  with \(0<\kappa + \frac{1}{\alpha }\le 2\)  there exists \(\varepsilon >0\)  small enough such that for for all \(q\ge \frac{8}{\varepsilon }\vee d\)  and for \(w_0 (x) = (1+|x|)\) 
there exists a positive random variable \(K(b) = K(b,\sigma ,\mathbf {W})\)  with \(K(b) \in L^q(\Omega )\)  and such that
\(\Vert T^{D\varphi ,\varphi }b\Vert _{\mathcal {C}^{\frac{1}{2}+\frac{3\varepsilon }{8}}_T \mathcal {C}^{\kappa + \frac{1}{\alpha }- \frac{\varepsilon }{8}}_{1+|\cdot |}} \le K(b) \Vert b\Vert _{\mathcal {C}^{\kappa }}. \) 
Hence, for \(\kappa + \frac{1}{\alpha }-\frac{\varepsilon }{8}+ \frac{1}{2} + \frac{3\varepsilon }{8}>1\) , namely for
\(\kappa > \frac{3}{2} + \frac{\varepsilon }{4},\) 
thanks to Theorem REF , there exists a unique solution, locally Lipschitz in the initial condition to Equation (REF ).
Finally we give a (standard) example that satisfies the conditions which are required for uniqueness of the solutions. We refer to [10]} and [7]} for the proofs of the following propositions.
Proposition 6.2 
Let \(H\in (0,1)\) . Let \(B,\tilde{B}\)  be two independent \(d\) -dimensional Brownian motions. We define the \(d\) -dimensional fractional Brownian motion of Hurst parameter \(H\)  as the process defined for all \(t\ge 0\)  by
\(B^H_t = \frac{1}{c_H}\left(\int _0^t (t-r)^{H-\frac{1}{2}} \mathop {}\!\mathrm {d}B_r + \int _0^{+\infty } (t+r)^{H-\frac{1}{2}} - r^{H-\frac{1}{2}} \mathop {}\!\mathrm {d}\tilde{B}_r\right),\) 
where
\(c_H = \sqrt{\frac{1}{2H}+\int _0^{+\infty } \big ((t+r)^{H-\frac{1}{2}} - r^{H-\frac{1}{2}}\big )^2 \mathop {}\!\mathrm {d}r }\) 
The fractional Brownian motion is a centered continuous Gaussian process with stationary increments and covariance
\(R_{B^H}(s,t) = \frac{1}{2}(t^{2H} + s^{2H} - |t-s|^{2H}).\) 
For \(\frac{1}{4}<H<\frac{1}{2}\)  it satisfies Assumption REF  with \(\rho = \frac{1}{2H}\) , Assumption REF  and Assumption REF  with \(\alpha = 2H\) .

Corollary 6.3 
Let \(H\in \left(\frac{1}{4},\frac{1}{2}\right)\) . Let \(B^H\)  be a \(d\) -dimensional fractional Brownian motion and let \(\mathbf {B}^H\)  the geometric rough path above \(B^H\) .
Let \(b \in \mathcal {C}^{\kappa }\)  with
\(\kappa > 0\vee \left(\frac{3}{2}-\frac{1}{2H}\right).\) 
Let \(\sigma \in C^{\infty }_b(\mathbb {R}^d;\mathbb {R}^{d\times d})\)  which satisfies the strong ellipticity condition there exists \(c>0\)  such that :
\( |\sigma (x) z|^2 \ge c|z|^2,\quad x,z\in \mathbb {R}^d.\) 
Then, there exists \(\mathcal {N}= \mathcal {N}(b) \in \mathcal {F}\)  such that \( \mathbb {P}(\mathcal {N}) = 0\)  and for all \(\omega \notin \mathcal {N}\)  and for all \(x\in \mathbb {R}^d\) ,
\(\mathop {}\!\mathrm {d}x_t = b(x_t) \mathop {}\!\mathrm {d}t + \sigma (x_t) \mathop {}\!\mathrm {d}\mathbf {B}^H_r,\quad x_0 = x,\quad t\in [0,T]\) 
admits a unique solution. Furthermore, for all \(R>0\)  and all all \(b,\tilde{b}\)  which satisfies the above conditions and all \(x_0,\tilde{x}_0 \in \mathbb {R}^d \)  with \(|x_0|,|\tilde{x}_0| \le R\) , there exists an almost surely positive and finite random variable \(K = K(\mathbf {B}^H,\sigma ,b,\tilde{b},R,T)\)  such that
\(\sup _{t\in [0,T]} |x_t - \tilde{x}_t| \le K (|x-\tilde{x}| - \Vert b - \tilde{b}\Vert _{\mathcal {C}^{\kappa }}),\) 
where \(x\)  (respectively \(\tilde{x}\) ) is the solution of the rough differential equation driven by the fractional Brownian motion with initial value \(x_0\)  (respectively \(\tilde{x}_0\) ) and coefficient \(b\)  and \(\sigma \)  (respectively \(\tilde{b}\)  and \(\sigma \) ).

One only has to use Theorem REF  and Proposition REF  and Theorem REF  to conclude.
Remark 6.4 Let us remark that we where not able to prove the usual continuity of the Itô map solution with respect to the driven rough path. Indeed, whenever \(B^{\varepsilon ,H}\)  is a smooth (say piecewise linear) approximation of the fractional Brownian motion, one does not necessarily have Proposition REF . In the same manner, the continuity with respect to the coefficient \(\sigma \)  is neither clear, since it appears in a deeply non-linear fashion manner in the averaged field.


appendix

Besov spaces
We first recall some of the Littlewood-Payley theory that is used in Besov spaces. There exist \(\psi ,\phi \)  two functions valued in \([0,1]\)  such that
1)
\(\psi ,\phi \in \mathcal {C}^{\infty }_0(\mathbb {R}^d) \)  with \(\textrm {supp}(\psi )\subset B(0,3/4)\)  and \(\textrm {supp}(\phi )\subset \mathcal {A}\) , where \(\mathcal {A}= \lbrace  \xi \in \mathbb {R}^d; 3/4\le |\xi |\le 8/3\rbrace \)  is an annulus in \(\mathbb {R}^d\) ,
2)
\(\forall \xi \in \mathbb {R}^d, \quad \psi (\xi ) + \sum _{j = 0}^{\infty } \phi (2^{-j}\xi ) = 1\) ,
3)
\(\textrm {supp}(\psi )\,\cap \, \textrm {supp}(\phi (2^{-j}\cdot ) = \emptyset \)  for any \(j\ge 1\) ,
4)
\(\textrm {supp}(\phi (2^{-j}\cdot ) \,\cap \, \textrm {supp}(\phi (2^{-k}\cdot ) = \emptyset \)  for any \(j,k\ge 0\)  such that \(|j-k|\ge 2\) .

We define the associated Littlewood-Payley blocks as
\(\Delta _{-1}u = \mathcal {F}^{-1}\left(\psi \mathcal {F}(u)\right) \quad \mbox{ and } \quad \Delta _j u = \mathcal {F}^{-1}\left(\phi (2^{-j}\cdot ) \mathcal {F}(u)\right), \quad \forall j\ge 0,\) 
for any \(u\in \mathcal {S}^{\prime }\) . We recall the Bernstein Lemmas in this context.
Lemme 1.1 Let \(k\in \mathbb {N}\) . There exists a constant \(C>0\)  such that, for all \(1\le q\le p\le \infty \)  and \(f\in L^p(\mathbb {R}^d)\) , we have
1)
\(\sup _{|\alpha | = k }\Vert  \partial ^{\alpha }\Delta _{j} f \Vert _{L^q(\mathbb {R}^d)} \le C^{k+1} 2^{k (j)_+ + d(1/q-1/p) }\Vert  \Delta _{j} f\Vert _{L^p(\mathbb {R}^d)}\) , for all \(j\in \mathbb {N}\cup \lbrace -1\rbrace \) ,
2)
\(C^{-k-1} 2^{jk} \Vert \Delta _j f\Vert _{L^p(\mathbb {R}^d)} \le \sup _{|\alpha | = k }\Vert  \partial ^{\alpha }\Delta _j f \Vert _{L^p(\mathbb {R}^d)} \le C^{k+1} 2^{jk} \Vert \Delta _j f\Vert _{L^p(\mathbb {R}^d)}\) , for all \(j\in \mathbb {N}\) .


The Besov spaces associated to the Littlewood-Payley blocks are defined as
\(B_{p,r}^{\alpha } : = \left\lbrace  u \in \mathcal {S}^{\prime }(\mathbb {R}^d); \Vert u\Vert _{B_{p,r}^\alpha } : = \left(\sum _{j = -1}^{+\infty } 2^{r\alpha j}\Vert \Delta _{j} u\Vert _{L^p(\mathbb {R}^d)}^r \right)^{1/r}<\infty \right\rbrace ,\) 

where \(s\in \mathbb {R}\)  and \(p,r\in [1,\infty ]\) . Those spaces are Banach spaces continuously embedded in \(\mathcal {S}^{\prime }\)  and they are of type \(p\wedge r\wedge 2\)  (see Corollary REF  below) when \(p,r\in (1,+\infty )\) .
Remark 1.2
1)
We have the following continuous embeddings
a)
for \(\tilde{\alpha }<\alpha \)  or \(\tilde{\alpha } = \alpha \)  and \(\tilde{r}\ge r\)
\(B_{p,r}^\alpha \hookrightarrow B_{p,\tilde{r}}^{\tilde{\alpha }},\) 
b)
for \(\tilde{p}\ge p\) ,
\(B_{p,r}^\alpha \hookrightarrow B_{\tilde{p},r}^{\alpha -d(1/p-1/\tilde{p})},\) 
c)
for \(p<\infty \) ,
\(B_{p,1}^{d/p} \hookrightarrow C^0_0,\) 
where \(C\)  is the space of uniformly continuous bounded functions.

2)
For \(p=r=\infty \)  and \(s\in \mathbb {R}^+/\mathbb {N}\) , we have that
\(\Vert u\Vert _{B_{\infty ,\infty }^s} \simeq \sum _{|\alpha |\le [s]}\left(\Vert \partial ^{\alpha }u\Vert _{L^{\infty }(\mathbb {R}^d)} + \sup _{\begin{array}{c}x,y\in \mathbb {R}^d\\x\ne y\end{array}}\frac{|\partial ^{\alpha }u(x)-\partial ^{\alpha }u(y)|}{|x-y|^{s-[s]}}\right).\) 
Thus, the spaces \(B_{\infty ,\infty }^\alpha \)  consist of \(\alpha \) -Hölder functions and are called Hölder-Besov spaces.
3)
For \(p = r\) , we have that
\(\Vert u\Vert _{B_{p,p}^\alpha } \simeq \Vert u\Vert _{H_p^\alpha (\mathbb {R}^d)},\) 
where
\(\Vert u\Vert _{H_p^\alpha (\mathbb {R}^d)} = \left\Vert  (1-\Delta )^{s/2} u\right\Vert _{L^p(\mathbb {R}^d)}\) 
and, thus, we recover the usual Sobolev spaces.
4)
Let \(k\in \mathbb {N}\) . It follows from Bernstein's lemma that, for any \(\alpha \in \mathbb {N}^d\)  such that \(|\alpha |\le k\) ,
\(\Vert \partial ^{\alpha }u\Vert _{B_{p,r}^\alpha } \lesssim \Vert u\Vert _{B_{p,r}^{s+k}}.\) 


It turns out that the space \(\mathcal {C}^{\infty }_0(\mathbb {R}^d)\)  is dense in \(B_{p,r}^\alpha \)  if and only if \(p,r<\infty \) . If we wish to build approximations of functions in the Hölder-Besov spaces, we can rely on following result.
Lemme 1.3 Let \(s\in \mathbb {R}\)  and \(u\in B_{\infty ,\infty }^\alpha \) . Then, the sequence \((u_n)_{n\ge 0}\)  defined by
\(u_n = \sum _{q = -1}^{n-1} \Delta _q u,\) 
converges to \(u\)  in \(B_{\infty ,\infty }^{\tilde{s}}\)  for any \(\tilde{s}<s\) . Moreover, we have, \(\forall n\in \mathbb {N}\)  and \(\forall \ell \in \mathbb {R}\) ,
\(\Vert u_n\Vert _{ B_{\infty ,\infty }^{\ell }} < +\infty .\) 

Finally, we recall the following results for pointwise product of functions in Besov spaces.
Lemme 1.4 
Let \(p \in [1,\infty [\) , \(r_1,r_2\in [1,\infty ]\)  and \(s_1,s_2\in \mathbb {R}^+\)  such that
\(s_1+s_2>0 \quad \textrm {and}\quad \max (s_1,s_2)<\frac{d}{p}.\) 
 Then, for any \(u\in B_{p,r_1}^{s_1}\)  and \(v\in B_{p,r_2}^{s_2}\) , we have
\(\Vert uv\Vert _{B_{p,r}^\alpha }\lesssim \Vert u\Vert _{B^{s_1}_{p,r_1}}\Vert v\Vert _{B^{s_2}_{p,r_2}},\) 
where
\(s = s_1 + s_2 - \frac{d}{p}\quad \textrm {and}\quad \frac{1}{r} = \max \left(1,\frac{1}{r_1}+\frac{1}{r_2}\right).\) 

We give a result concerning Fourier multiplier.
Lemme 1.5 
Let \(m:\mathbb {R}^d\rightarrow \mathbb {R}\)  be a smooth function such that there, \(\gamma \in \mathbb {N}^d\)  and \(p\ge 0\) ,
\(\partial ^{\gamma }m(\xi ) \le |\xi |^{-|\gamma |-p},\; \xi \in \mathbb {R}^d\backslash \lbrace 0\rbrace ,\) 
and, for any \(j\) , define the function \(\mathsf {m}_j\)  given by
\(\mathsf {m}_j = \mathcal {F}^{-1}(m(\xi )\varrho (2^{-j}\xi )),\) 
where \(\varrho \)  is a compactly supported function such that \(\mathrm {supp}(\varrho )\subset 2\mathcal {A}\)  and \(\varrho _{|\mathcal {A}} \equiv 1\) . Then, we have
\(\Vert \mathsf {m}_j \Vert _{L^1(\mathbb {R}^d)}\lesssim _{\varrho ,d} 2^{-jp}.\) 

It turns out that, by a change of variable,
\(\Vert \mathsf {m}_j\Vert _{L^1(\mathbb {R}^d)} = \Vert \mathcal {F}^{-1}(m(2^j\xi )\varrho (\xi ))\Vert _{L^1(\mathbb {R}^d)}.\) 
Furthermore, we have, by integration by parts and Leibniz's formula,
\(\mathcal {F}^{-1}(m(2^j\xi )\varrho (\xi ))(x) &= (1+|x|^2)^{-d}(2\pi )^{-d/2}\int _{\mathbb {R}^d}e^{i\xi \cdot x} (1-\Delta )^{d}(m(2^j\xi )\varrho (\xi )) \mathop {}\!\mathrm {d}\xi \\ &=(1+|x|^2)^{-d}(2\pi )^{-d/2}\sum _{\begin{array}{c}\alpha ,\beta \in \mathbb {N}^d\\ \beta \le \alpha ,\; |\alpha |\le 2d\end{array}}C_{\alpha ,\beta } 2^{j|\alpha |}\int _{\mathbb {R}^d}e^{i\xi \cdot x} \partial ^{\alpha }_{\xi }m(2^j\xi )\partial ^{\alpha -\beta }\varrho (\xi ) \mathop {}\!\mathrm {d}\xi .\) 

We can see that, by (REF )
\(\left|\int _{\mathbb {R}^d}e^{i\xi \cdot x} \partial ^{\alpha }_{\xi }m(2^j\xi )\partial ^{\alpha -\beta }\varrho (\xi ) \mathop {}\!\mathrm {d}\xi \right| \lesssim _{\mathcal {A}} 2^{-j|\alpha |-jp},\) 
which leads to the estimate
\(\Vert \mathsf {m}_j\Vert _{L^1(\mathbb {R}^d)} \lesssim 2^{-jp}\) 

Young inequality
Let us recall the following Young inequality for Kernel (See [29]}, Theorem 0.3.1).
Theorem 2.1 
Let \(K:\mathbb {R}^d\times \mathbb {R}^d \mapsto \mathbb {R}\)  be a measurable function such that for some \(p\in [1,+\infty ]\) ,
\(\sup _{x\in \mathbb {R}^d} \Vert K(x,\cdot )\Vert _{L^p(\mathbb {R}^d)} \le C \quad \text{ and } \quad \sup _{y\in \mathbb {R}^d} \Vert K(\cdot ,y)\Vert _{L^p(\mathbb {R}^d)} \le C.\) 
Then \(p,q,r\in [1,+\infty ]\)  with \(\frac{1}{p}+\frac{1}{q} = 1+\frac{1}{r}\)  and for all \(f \in L^{q}(\mathbb {R}^d)\) , we have
\(\left(\int _{\mathbb {R}^d} \left|\int _{\mathbb {R}^d} K(x,y) f(y)\mathop {}\!\mathrm {d}y \right|^r \mathop {}\!\mathrm {d}x\right)^{\frac{1}{r}} \le C \Vert f\Vert _{L^q(\mathbb {R}^d)}.\) 


Burkolder Davis Gundy inequality in Lebesgue spaces
In this section, we recall some standard results about martingales in Banach spaces, and the corresponding inequalities. We refer to Pisier [30]} chapter 4 and to Pinelis [31]} and Veraar and coauthors [1]}, [2]} for proofs.
The aim of this section is to give an almost self-contain proof of Burkolder-Davis-Gundy inequality in infinite dimensional Banach spaces, especially in \(L^p(\mathbb {R}^d)\)  spaces. The idea of the proof is to use Kahane-Khinchine inequality, and UMD properties vie smoothness of spaces.
Discrete Fourier transform and Kahane-Khinchine inequality
We recall some of standard feature about the Fourier transform on the hypercube.
Let \(\Omega _n = \lbrace -1,1\rbrace ^n\) . We endowed \(\Omega _n\)  with the uniform measure \(\mu _n\) , such that for a function \(f : \Omega _n \mapsto \mathbb {R}\)  we have
\(\int _{\Omega _n} f \mathop {}\!\mathrm {d}\mu _n = \mathbb {E}[f(X)] = \frac{1}{2^n} \sum _{x\in \Omega _n} f(x)\) 
where \(X=(X_1,\cdots ,X_n)\)  is a vector of i.i.d. random variables such that \(\mathbb {P}(X_1 = 1) = \mathbb {P}(X_1 = -1) = \frac{1}{2}\) .
Remark that for any \(Y\in L^2(\Omega _n,\mathcal {P}(\Omega _n),\mu _n)\)  a unique function \(f : \Omega _n \mapsto \mathbb {R}\)  exists such that \(Y = f(X)\) . Hence \(L^2(\Omega _n,\mathcal {P}(\Omega _n),\mu _n)\)  is a finite dimensional vector space of dimension \(2^n\) . For \(Y_1 = f_1(X),Y_2=f_2(X) \in L^2(\Omega _n,\mathcal {P}(\Omega _n),\mu _n)\)  let us write the standard inner product as
\(\langle Y_1, Y_2\rangle = \mathbb {E}[f_1(X) f_2(X)].\) 
Furthermore let us define for \(S\subset \lbrace 1,\cdots ,n\rbrace \) , we define
\(X_S = \prod _{l\in S} X_l,\quad X_{\emptyset } = 1.\) 
Remark that \(\langle X_S , X_S \rangle = 1\)  and for \(S^{\prime }\ne S\)  we have \(\langle X_S , X_S^{\prime } \rangle = 0\) . Hence \(X_S\)  is a orthogonal basis of \(L^2(\Omega _n,\mathcal {P}(\Omega _n),\mu _n)\) . Furthermore for \(Y \in L^2(\Omega _n,\mathcal {P}(\Omega _n),\mu _n)\)  and \(S\subset \lbrace 1,\cdots ,n\rbrace \) , we define
\(\hat{Y}(S) = \langle Y , X_S \rangle = \mathbb {E}[Y X_S] ,\) 
such that
\(Y= \sum _{S\subset \lbrace 1,\cdots ,n\rbrace } \hat{Y}(S) X_S,\) 
and
\(\mathbb {E}[YZ]=\sum _{S\subset \lbrace 1,\cdots ,n\rbrace }\hat{Y}(S)\hat{Z}(S).\) 
Note that we have the two following identities :
\(\mathbb {E}[Y^2]= \sum _{§\subset \lbrace 1,\cdots ,n\rbrace } \hat{Y}(S)^2\) 
and
\(\mathbb {E}[Y] = \hat{Y}(\emptyset ).\) 
Theorem 3.1 (Khinchine-Kahane Inequality, via Szarek's proof, [34]} Theorem 5.20) 
Let \((B,|\cdot |_B)\)  be a normed vector space. Then for any \(p\in [1,+\infty ]\) , there exists a constant \(C_p\)  such that for any \(n\ge 2\)  and for any \(b_1,\cdots ,b_n \in B\) , we have
\(\mathbb {E}\left[\left|\sum _{i=1}^n b_i X_i\right|^p_B\right] \le C_p \mathbb {E}\left[\left|\sum _{i=1}^n b_i X_i\right|_B\right],\) 
where \(X=(X_1,\cdots ,X_n)\)  is an iid random vector with \(\mathbb {P}(X_1 = 1) = \mathbb {P}(X_1 = -1)= \frac{1}{2}\) .

Let \(k\ge 1\)  such that \(k\le p < k+1\) .
\(\mathbb {E}\left[\left|\sum _{i=1}^n b_i X_i\right|^p_B\right]^{\frac{1}{p}}\le \mathbb {E}\left[\left|\sum _{i=1}^n b_i X_i\right|^{k+1}_B\right]^{\frac{1}{k+1}}\\\le C_{k+1} \mathbb {E}\left[\left|\sum _{i=1}^n b_i X_i\right|_B\right].\) 
It is enough to prove the Theorem for \(k+1\) .
Let us define
\(Z = \left|\sum _{j=1}^n b_j X_j\right|_B,\) 
Let us define for \(i\in \lbrace 1,\cdots ,n\rbrace \) ,
\(X^{(i)}=(X^{(i)}_1,\cdots X^{(i)}_{n})=(X_1,\cdots , X_{i-1},-X_i,X_{i+1},\cdots ,X_n),\) 
such that \(X^{(i)} \overset{(d)}{=} X\) , and
\(Z^{(i)} = \left| \sum _{j=1,j\ne i}^n b_j X^{(i)}_j\right|_B\) 
and finally
\(\bar{Z} = \sum _{i=1}^n Z^{(i)}.\) 
The core of the proof is then the following : we will give some lower and upper bounds for \(\mathbb {E}[Z^k \bar{Z}]\)  with respect to \(\mathbb {E}[Z]\) , \(\mathbb {E}[Z^{k+1}]\) . To do so, we will give a direct lower bound to \(\bar{Z}\) . For the upper bound, we will dramatically use the Fourier transform.
Let us focus first on the lower bound. We have
\(\bar{Z}=\sum _{i=1}^n\left|\sum _{j=1}^n X^{(i)}_j b_j\right|_B \\\ge \left|\sum _{i=1}^n \sum _{j=1}^n X^{(i)}_j b_j\right|_B \\= (n - 2) Z.\) 
Since \(Z^k\)  is non negative, we have
\((n - 2) \mathbb {E}[Z^{k+1}] \le \mathbb {E}[Z^k \bar{Z}].\) 
Let us now focus on the upper bound. First note that if we define for \(S\subset \lbrace 1,\cdots ,n\rbrace \)  and \(i\in \lbrace 1,\cdots ,n\rbrace \) ,
\(X_S^{(i)} = \prod _{l\in S} X^{(i)}_l,\quad X_{\emptyset }^{(i)} = 1,\) 
then \((X_S^{(i)})_{S\subset \lbrace 1,\cdots ,n\rbrace }\)  is a orthonormal basis of \(L^2\)  and furthermore note that \((Z^{(i)},X^{(i)}_S)\)  has the same distribution as \((Z,X_S)\)  for all \(i\)  and all \(S\) .
Note also that
\(X_S^{(i)} ={\left\lbrace \begin{array}{ll}X_S & \text{if } i \notin S\\-X_S & \text{if } i \in S\end{array}\right.}.\) 
Hence,
\(\hat{\bar{Z}}(S) = & \sum _{i=1}^n \mathbb {E}[Z^{(i)} X_S] \\=& \sum _{i\notin S} \mathbb {E}[Z^{(i)} X^{(i)}_S] - \sum _{i\in S}\mathbb {E}[Z^{(i)} X^{(i)}_S] \\=& (n-2|S|) \mathbb {E}[Z X_S] \\=& (n-2|S|) \hat{Z}(S).\) 

Not also that if there exists \(m\)  such that \(|S|=2m+1\) , since \(X\overset{(d)}{=}-X\)  we have
\(\hat{Z}(S)=\mathbb {E}[Z X_S ] = \left[\left|\sum _{j=1}^{n} -X_j b_j \right|_{B} \prod _{l\in S}(-X_l)\right] = -\hat{Z}(S),\) 
and
\(\hat{Z}(S) = 0.\) 
Hence we have
\(\mathbb {E}[Z^k \bar{Z}]=&\sum _{S\subset \lbrace 1,\cdots ,n\rbrace } \widehat{Z^k}(S)\hat{\bar{Z}}(S)\\=&\sum _{S\subset \lbrace 1,\cdots ,n\rbrace } (n-2|S|)\widehat{Z^{k}}(S)\hat{Z}(S) \\=&n\widehat{Z^k}(\emptyset )\hat{Z}(\emptyset ) + \sum _{\begin{array}{c}S\subset \lbrace 1,\cdots ,n\rbrace \\|S|=2m\\S\ne \emptyset \end{array}} (n-2|S|) \widehat{Z^k}(S)\hat{Z}(S) \\\le &4\mathbb {E}[Z^k]\mathbb {E}[Z] + (n-4) \sum _{S\subset \lbrace 1,\cdots ,n\rbrace } \widehat{Z^k}(S) \hat{Z}(S) \\=&4\mathbb {E}[Z^k]\mathbb {E}[Z] + (n-4) \mathbb {E}[Z^{k+1}].\) 
Putting upper and lower bounds together, we get
\(\mathbb {E}[Z^{k+1}] \le 2 \mathbb {E}[Z^k]\mathbb {E}[Z].\) 
By a direct induction, we have
\(\mathbb {E}[Z^{k}]^{\frac{1}{k}} \le 2^{1-\frac{1}{k}} \mathbb {E}[Z],\) 
which is the wanted bound with \(C_k = 2^{1-\frac{1}{k}}\)

UMD spaces
Definition 3.2 Let \((B,|\cdot |_B)\)  ]
be a Banach space and \(p\in (1,+\infty )\) . We say that \(B\)  as the Unconditionnal Martingale Difference-\(p\)  (\(\text{UMD}_p\)  ) property if there exists a constant
\(K_p(B)>0\)  such that for any filtered probability space \((\Omega ,\mathcal {F}, (\mathcal {F}_n)_{n\ge 0},\mathbb {P})\)  and any \(B\)  value martingale \((M_n)_{n\ge 0}\)  with \(p\)  moments, any \(N\ge 1\)  and any \(\varepsilon _1,\cdots ,\varepsilon _N \in \lbrace -1,1\rbrace \) ,
\(\mathbb {E}\left[\left|\sum _{k=0}^{N-1} \varepsilon _k (M_{k+1}-M_k)\right|^p_B\right] \le K_p(B)^p \mathbb {E}[|M_N-M_0|_B^p]\) 
We say that \(B\)  as the UMD property if \(B\)  is \(\text{UMD}_p\)  for all \(p\in (0,+\infty )\) .

The following theorem is a fundamental tool in the theory of UMD spaces. We refer to [35]} for a proof of it (and much more).
Theorem 3.3 Let \((B,|\cdot |_B)\)  be a Banach space with the \(\text{UMD}_p\)  property. Then \((B,|\cdot |_B)\)  has the UMD property.

The following proposition emphasize the fact that UMD spaces and inequaltiy for martingales are strongly linked :
Proposition 3.4 The finite dimensional vector space \(\mathbb {R}^d\)  endowed with its standard euclidean norm has the UMD property.

The proof relies on the classical discrete Burkolder Davis Gundy (BDG) inequality. Indeed let \((M_n)_n\)  be an \(L^p\)  martingale in \(\mathbb {R}^d\) . Let \(\varepsilon _1,\cdots ,\varepsilon _N \in \lbrace -1,1\rbrace \) , \(M^\varepsilon _0 = 0\)  and for \(N\in \lbrace 1,\cdots ,N\rbrace \)
\(M^\varepsilon _n = \sum _{k=0}^{n-1} \varepsilon _k (M_{k+1} - M_k).\) 
Then \((M^\varepsilon _n)_{n\in \lbrace 0,\cdots ,N\rbrace }\)  is a martingale, and we have
\(\mathbb {E}\left[\left|M^\varepsilon _N\right|^p\right]\lesssim \mathbb {E}\left[\left(\sum _{k=0}^{N-1}\left|M^\varepsilon _{k+1}-M^\varepsilon _{k+1}\right|^2\right)^{\frac{p}{2}}\right]\\ =\mathbb {E}\left[\left(\sum _{k=0}^{N-1}\left|M_{k+1}-M_{k+1}\right|^2\right)^{\frac{p}{2}}\right]\lesssim \mathbb {E}[|M_N - M_0|^p],\) 
were we have applied twice the BDG inequality.
Theorem 3.5 
Let \(p\in (1,+\infty )\) , let \((B,|\cdot |_B)\)  be an Banach space with the UMD and let \((A,\mathcal {A},\mu )\)  be a measured space. The space \(L^p(A;B)\)  is a UMD space.

The proof follows easily from the previous results. Indeed, it is enough to prove that \(L^p(A;B)\)  has the \(\text{UMD}_p\)  property. Let \((M_n)_{n}\)  be a \(L^p\)  martingale with value in \(L^p(A;B)\) . Remark that for \(\mu \) -almost all \(x \in A\) , \(\big (M_n(x)\big )_n\)  is a \(B\)  value martingale. Furthermore, thanks to the Fubini property, we have
\(\mathbb {E}\left[\Vert M_n\Vert _{L^p(A;B)}^p\right] = \int _A \mathbb {E}[|M_n(x)|_B^p] \mathop {}\!\mathrm {d}\mu (x).\) 
Hence for \(\mu \) -almost all \(x\) , \(\mathbb {E}[|M_n(x)|_B^p]<+\infty \)  and \(\big (M_n(x)\big )_n\)  is a \(L^p\)  martingale with value in \(B\) . Hence for all \(\varepsilon _1,\cdots ,\varepsilon _n \in \lbrace -1,1\rbrace \) , we have for almost all \(x\in A\)  and all \(N\ge 1\)
\(\mathbb {E}[|M_N^\varepsilon (x)|^p_B] \le K_p(B)^p \mathbb {E}[|M_N(x) - M_0(x)|^p_B].\) 
Hence, we have
\(\mathbb {E}[\Vert M^\varepsilon _N\Vert _{L^p(A;B)}^p] = \int _A \mathbb {E}[ |M^\varepsilon _N(x)|^p_B ] \mathop {}\!\mathrm {d}\mu (x) \\ \le K_p(B)^p \int _A \mathbb {E}[ |M_N(x)- M_0(x)|^p_B ] \mathop {}\!\mathrm {d}\mu (x) = K_p(B)^p \mathbb {E}[\Vert M_N - M_0\Vert _{L^p(A;B)}^p].\) 
Hence, \(L^p(A;B)\)  is \(\text{UMD}_p\)  and then UMD.
Corollary 3.6 
Let \(p,q \in (1,+\infty )\)  and let \(\alpha \in \mathbb {R}\) . Then \(B^\alpha _{p,q}(\mathbb {R}^d,\mathbb {R})\)  is UMD.

Is previously it is enough to prove that \(B^\alpha _{p,q}\)  is \(\text{UMD}_q\) . Furthermore, since \(L^p(\mathbb {R}^d;\mathbb {R})\)  is UMD we have for al \(N\in \mathbb {N}\)  and all \(\varepsilon _1,\cdots ,\varepsilon _N \in \lbrace -1,1\rbrace \) ,
\(\mathbb {E}[\Vert M^\varepsilon _N\Vert _{B^\alpha _{p,q}}^q=\sum _{j\ge -1} 2^{\alpha j q} \mathbb {E}[ \Vert \Delta _j M^\varepsilon _N\Vert ^q_{L^p(\mathbb {R}^d;\mathbb {R})}]= K_q(L^p(\mathbb {R}^d;\mathbb {R}))^q \mathbb {E}[\Vert M_N - M_0\Vert _{B^\alpha _{p,q}}^q],\) 
and the result follows easily.

Banach space, smoothness and type.
Definition 3.7 Let \(p\in (1,2]\)  and \(B,|\cdot |_B)\)  be a Banach space. We say that \(B\)  is of type \(p\)  if there exists a constant \(M_p\)  such that for every \(N\ge 0\) , every iid sequence \(X_1\cdots ,X_n\)  with \(\mathbb {P}(X_1 = 1) = \mathbb {P}(X_1 = -1) = \frac{1}{2}\)  and every \(b_1,\cdots ,b_N \in B\) , we have
\(\mathbb {E}\left[\left|\sum _{k=1}^N X_k b_k\right|_B\right]\le M_p\left(\sum _{k=1}^N|b_k|_B^p\right)^{\frac{1}{p}}.\) 

Let us first give an example :
Proposition 3.8 Let \(\mathbb {R}^d\)  with its Euclidean norm. Then \(\mathbb {R}^d\)  is of type 2. More generally any Hilbert space is of type 2.

The proof is quite straightforwerd. Let us denote by \(\langle \cdot ,\cdot ,\rangle \)  the inner product. We have for \(N\ge 1\) , \(b_1,\cdots ,b_N \in \mathbb {R}^d\)  and \(X_1,\cdots ,X_n\)  a sequence of iid random variables with \(\mathbb {P}(X_1 = 1 ) = \mathbb {P}(X_1 = -1) = \frac{1}{2}\) ,
\(\mathbb {E}\left[\left|\sum _{k=1}^N X_k b_k\right|^2\right] = \sum _{k,l=1}^N \mathbb {E}\left[X_k X_l\right] \langle b_k , b_l \rangle =&\sum _{k=1}^N |b_k|^2.\) 

Applying the Khinchine-Kahane inequality, we have the result.
Theorem 3.9 
Let \((A,\mathcal {A},\mu )\)  be a measured space and let \((B,|\cdot |_B)\)  be a type \(p\)  Banach space with \(p\in (1,2]\) . Let \(q\in [1,+\infty )\) . Then \(L^q(A;B)\)  is of type \(p\wedge q\) .

Take \(N\in \mathbb {N}\) , \(b_1,\cdots ,b_N \in L^q(A;B)\)  and \(X_1,\cdots ,X_N\)  a sequence of iid random variables with \(\mathbb {P}(X_1 = 1 ) = \mathbb {P}(X_1 = -1) = \frac{1}{2}\) .
Applying twice the Khinchine-Kahane inequality, we get
\(\mathbb {E}\left[\left\Vert \sum _{k=1}^n X_k b_k\right\Vert _{L^q(A;B)}\right]\lesssim &\mathbb {E}\left[\left|\sum _{k=1}^n X_k b_k\right|^q\right]^{\frac{1}{q}} \\& \lesssim \left(\int _A \mathbb {E}\left[\left|\sum _{k=1}^n X_k b_k(x)\right|_B^q\right] \mathop {}\!\mathrm {d}\mu (x)\right)^{\frac{1}{q}} \\& \lesssim \left(\int _A \mathbb {E}\left[\left|\sum _{k=1}^n X_k b_k(x)\right|_B\right]^q \mathop {}\!\mathrm {d}\mu (x)\right)^{\frac{1}{q}} \\& \lesssim \left(\int _A \left(\sum _{k=1}^n |b_k(x)|_B^p\right)^{\frac{q}{p}} \mathop {}\!\mathrm {d}\mu (x)\right)^{\frac{1}{q}},\\\) 

where we have used the fact that \((B,|\cdot |_B)\)  is of type \(p\)  in the last inequality.
Now, let us suppose that \(q\le p\) . We have for every \(N\ge 1\)  and \(y_1,\cdots ,y_N \in \mathbb {N}\) ,
\(\left(\sum _{k=1}^N |y_i|^p \right)^{\frac{1}{p}} \le \left(\sum _{k=1}^N |y_i|^q\right)^{\frac{1}{q}}.\) 
Finally we have
\(\mathbb {E}\left[\left\Vert \sum _{k=1}^N X_k b_k\right\Vert _{L^q(A;B)}\right] \lesssim \left(\int _A \sum _{k=1}^N |b_k(x)|_B^q \mathop {}\!\mathrm {d}\mu (x)\right)^{\frac{1}{q}} \lesssim \left(\sum _{k=1}^N \Vert b_k\Vert _{L^{q}(A;B)}^q\right)^{\frac{1}{q}}.\) 
Hence, in that case we have proved \(L^q(A;B)\)  which is of \(q\)  type.
Let us suppose that \(p\le q\) . Note that we have
\(\mathbb {E}\left[\left\Vert \sum _{k=1}^n X_k b_k\right\Vert _{L^q(A;B)}\right]\lesssim &\left(\left\lbrace \int _A \left(\sum _{k=1}^n |b_k(x)|_B^p\right)^{\frac{q}{p}} \mathop {}\!\mathrm {d}\mu (x)\right\rbrace ^{\frac{p}{q}}\right)^{\frac{1}{p}} \\\lesssim &\left( \sum _{k=1}^N \left(\int _A |b_k(x)|_B^q \mathop {}\!\mathrm {d}\mu (x) \right)^{\frac{p}{q}} \right)^{\frac{1}{p}}\\\lesssim &\left( \sum _{k=1}^N \Vert b_k\Vert _{L^q(A;B)}^p \right)^{\frac{1}{p}}.\) 

In that case \(L^q(A;B)\)  if of \(q\)  type, which ends the proof.
Corollary 3.10 
Let \(1\le p,q < + \infty \)  and \(\alpha \in \mathbb {R}\) . Then \(B^{\alpha }_{p,q}(\mathbb {R}^d;\mathbb {R})\)  is of type \(p\wedge q \wedge 2\) .

The proof follows easily when we see that
\(\Vert f\Vert _{B^\alpha _{p,q}} = \Vert (2^{j\alpha } f)_{j\ge -1}\Vert _{\ell ^q(L^p(\mathbb {R}^d;\mathbb {R}))}.\) 
By the previous result, \(L^p(\mathbb {R}^d;\mathbb {R})\)  is of type \(p \wedge 2\)  and \(\ell ^q(L^p(\mathbb {R}^d;\mathbb {R}))\)  is of type \(q \wedge p \wedge 2\) .

Burkolder-Davis-Gundy in Banach spaces
We now have all the tools to ennounce and proof the following theorem. One can consult [30]} chapter 4 for more details.
Theorem 3.11 Let \((B,|\cdot |_B)\)  be a UMD Banach space of type \(p \in (1,2]\) . Then for any \(r\in (1,+\infty )\)  a constant \(C_r>0\)  exists such that for any \(N\ge 1\)  and any \(L^r\)  martingale with value in \(B\)
\(\mathbb {E}[|M_N-M_0|^r] \le C_r^r \mathbb {E}\left[\left(\sum _{k=0}^{N-1} | M_{k+1}-M_k|_B^{\frac{r}{p}}\right)^{\frac{p}{r}}\right].\) 

Let \((M_n)_n\)  as in the theorem and \(N\ge 1\) . Let \((X_1,\cdots ,X_N)\)  be an iid random vector independent of \(M\)  with \(\mathbb {P}(X_1 = 1) = \mathbb {P}(X_1 = -1) = \frac{1}{2}\) .
First, remark that thanks to the UMD property, we have for all \(\varepsilon _1,\cdots ,\varepsilon _n \in \lbrace -1,1\rbrace \) ,
\(\mathbb {E}[|M_N - M_0|_B^r] \lesssim \mathbb {E}[|M^\varepsilon _N|_B^r].\) 
Hence, we have
\(\mathbb {E}[|M_N - M_0|_B^r]\lesssim &\mathbb {E}\left[\left|\sum _{k=0}^{N-1} X_k(M_{k+1} - M_k)\right|_B^r\right] & (\text{UMD}) \\\lesssim &\mathbb {E}\left[ \mathbb {E}\left[\left|\sum _{k=0}^{N-1} X_k(M_{k+1} - M_k)\right|_B^r\Big |(M_n)_{n\in \lbrace 0,\cdots ,N}\right]\right]\\\lesssim &\mathbb {E}\left[ \mathbb {E}\left[\left|\sum _{k=0}^{N-1} X_k(M_{k+1} - M_k)\right|_B\Big |(M_n)_{n\in \lbrace 0,\cdots ,N}\right]^r\right] & (\text{Khinchine-Kahane})\\\lesssim &\mathbb {E}\left[ \left(\sum _{k=0}^{N-1} |M_{k+1} - M_k)|^p_B\right)^{\frac{r}{p}}\right] & (\text{type}-p).\) 
Using Corollaries REF  and REF  and Theorems REF  and REF , we have the following useful corollary :
Corollary 3.12 Let \(p,q\in (1,+\infty )\)  and \(\alpha \in \mathbb {R}\) . The previous Inequality REF  holds for \(L^p(\mathbb {R}^d;\mathbb {R})\)  (respectively \(B^{\alpha }_{p,q}(\mathbb {R}^d;\mathbb {R})\) ) with \(p\wedge 2\)  instead of \(p\)  (respectively with \(p\wedge 2 \wedge q\)  instead of \(p\) ).

Remark 3.13 Using the fact that \(F^\alpha _{p,q}(\mathbb {R}^d;\mathbb {R})\)  is endowed in \(L^{q}(\ell ^p)\) , one could prove excatly the same theorem for Triebel-Lizorkin spaces.


The multi-parameter and multi-dimensionnal Garsia-Rodemich-Rumsey inequality.
Let us recall the standard Garsia-Rodemich-Rumsey inequality [37]}.
Theorem 4.1 
Let \(p\ge 1\) , \(\alpha >p^{-1}\)  and \(f\in \mathcal {C}([0,1])\) . Then, there exists a constant \(C_{\alpha ,p}>0\)  such that
\(|f(t)-f(s)| \le C_{\alpha ,p} \kappa _f |s-t|^{\alpha -1/p}\) 
for all \(t,s\in [0,1]^2\) , where
\(\kappa _f = \left(\int _{[0,1]^2} \frac{|f(u) - f(v)|^p}{|u-v|^{\alpha p + 1}}dudv\right)^{1/p}.\) 

We can extend the previous result to the case \(f\in \mathcal {C}([0,1]\times B(0,R) )\)  where \(B(0,R)\subset \mathbb {R}^d\)  is a closed ball of radius \(R>0\) . By denoting, for any \(x,y\in [0,1]^d\)  and \(s,t\in [0,1]\) ,
\(\square _{(s,x)} f(t,y) = f(s,x) - f(s,y) - f(t,x) + f(t,y),\) 
we have the following result.
Corollary 4.2 
Let \(p\ge 1\) , \(\alpha _1>1/p\) , \(\alpha _2>d/p\)  and \(f\in \mathcal {C}([0,1]\times B(0,R) )\) . Then, there exists a constant \(C_{d,\alpha _1,\alpha _2,p}>0\)  such that
\(|\square _{(s,x)} f(t,y)|^p \le C_{d,\alpha _1,\alpha _2,p} \kappa _f |s-t|^{\alpha _1p-1}|x-y|^{\alpha _2 p - d}\) 
for all \(t,s\in [0,1]\)  and \(x,y\in B(0,R)\) , where
\(\kappa _f =\int _{[0,1]^{2}\times B(0,R)^2} \frac{|\square _{(u,z)} f(v,w)|^p}{|u-v|^{\alpha _1p + 1}|z-w|^{\alpha _2p+d}}dudv dzdw.\) 

We essentially use the arguments from [38]} and [39]}. Let \(\lbrace (E_j,d_j)\rbrace _{1\le j \le m}\)  be a family of separable metric space each endowed with a finite doubling measure \(\upsilon _j\)  defined on the Borel sets of \(E_j\)  (we essentially need that Lebesgue's differentiation theorem holds) and \(\Psi \)  a positive increasing convex function such that \(\Psi (0) = 0\)  (and, thus, \(\Psi ^{-1}\)  is a positive increasing concave function). We define \(\sigma _j(r) = \min _{x\in E_j} \upsilon _j(B(x,r))\)  the volume (under \(\upsilon _j\) ) of the smallest ball of radius \(r>0\)  in \(E_j\) . For any metric space \((E,d)\)  endowed with a measure \(\upsilon \)  defined on the Borel sets of \(E\)  and for any scalar-valued function \(f\in \mathcal {C}(E)\) , we define
\(\bar{f}(A) : = \upsilon (A)^{-1} \int _{A} f(x) \upsilon (dx),\) 
for any Borel set \(A\in E\) . For any \((E,d,\upsilon )\in \lbrace (E_j,d_j,\upsilon _j)\rbrace _{1\le j\le m}\) , we have, by Lebesgue's differentiation theorem
\(\bar{f}(B(x,r)) \underset{r\rightarrow 0}{\rightarrow } f(x).\) 
We denote in the following \(\mathcal {E}= \bigotimes _{j = 1}^m E_j\)  the product space that is can be made metric by considering \(d(x,y) = \left(\sum _{j=1}^m d(x_j,y_j)^2\right)^{1/2}\) . It can also be made measurable by considering the product measure \(\upsilon (A) = \prod _{j = 1}^m \upsilon (A_j)\)  for any Borel set \(A = \prod _{j = 1}^m A_j\) . For any \(x = (x_1,x_2,\ldots ,x_m),y = (y_1,y_2,\ldots ,y_m)\in \mathcal {E}\) , we denote \(x^{\prime } = (x_1,\ldots ,x_{m-1})\)  (for \(m\ge 2\) ) and define, for any \(1\le k\le m\) ,
\(V_{k,y}x : = (x_1,\ldots ,x_{k-1},y_k,x_{k+1},\ldots ,x_m).\) 
Furthermore, for any function \(f\)  defined on \(\mathcal {E}\) , we denote \(V_{k,y}f(x) = f(V_{k,y}x)\) . We proceed to define the joint increment of \(f\)  as
\(\square _y f(x) : = \prod _{j = 1}^m (\textrm {id} - V_{k,y})f(x).\) 
From the previous definitions, we can see that, for any \(x,y\in \mathcal {E}\)  and \(f\)  defined on \(\mathcal {E}\)
\(\square _y f(x) = \square _{y^{\prime }} f(x^{\prime },x_m) - \square _{y^{\prime }} f(x^{\prime },y_m).\) 
We can now state an intermediate result.
Lemme 4.3 
Let \(f\in \mathcal {C}(\mathcal {E})\)  be such that
\(\kappa _f : = \int _{\mathcal {E}}\int _{\mathcal {E}} \Psi \left(\frac{|\square _yf(x)|}{\prod _{j=1}^m d_j(x_i,y_j)} \right)\upsilon (dx)\upsilon (dy)< +\infty .\) 
Then the following inequality holds for any \(x,y\in \mathcal {E}\)
\(|\square _yf(x)| \le 18^m \int _0^{d_1(x_1,y_1)/2}\ldots \int _0^{d_m(x_m,y_m)/2}\Psi ^{-1}\left(\frac{\kappa _f}{\prod _{j = 1}^m \sigma _j(r_j)^2}\right)\mathop {}\!\mathrm {d}r_1\ldots \mathop {}\!\mathrm {d}r_m.\) 

We proceed by induction; the case \(m = 1\)  being the classical Garsia-Rodemich-Rumsey. We assume that the result holds for a certain \(m-1\ge 1\) . For any \(x,y\in \mathcal {E}\) , we denote \(g_{x^{\prime },y^{\prime }}(z) =\square _{y^{\prime }} f(x^{\prime },z)\) . For any Borel sets \(A_1,A_2\subset E_m\) , we have
\(\left|\bar{g}_{x^{\prime },y^{\prime }}(A_1) - \bar{g}_{x^{\prime },y^{\prime }}(A_2)\right| &\le \int _{A_1}\int _{A_2} \left|g_{x^{\prime },y^{\prime }}(x_m) - g_{x^{\prime },y^{\prime }}(y_m)\right|\frac{\upsilon _m(dx_m)\upsilon _m(dy_m)}{\upsilon _m(A_1)\upsilon _m(A_2)}\\ &\le d_m(A_1,A_2) \int _{A_1}\int _{A_2} \left|\frac{g_{x^{\prime },y^{\prime }}(x_m) - g_{x^{\prime },y^{\prime }}(y_m)}{d_m(x_m,y_m)}\right|\frac{\upsilon _m(dx_m)\upsilon _m(dy_m)}{\upsilon _m(A_1)\upsilon _m(A_2)},\) 

where \(d_m(A_1,A_2) = \sup _{x_m\in A_1,y_m\in A_2} d_m(x_m,y_m)\) .
By the induction hypothesis, we have that
\(\frac{|g_{x^{\prime },y^{\prime }}(x_m) - g_{x^{\prime },y^{\prime }}(y_m)|}{d_m(x_m,y_m)} = \left|\square _{y^{\prime }} \delta _{x_m,y_m}f(x^{\prime })\right|\\ \le 18^{m-1} \int _{D}\Psi ^{-1}\left(\frac{\kappa _{\delta _{x_m,y_m}f}}{\prod _{j = 1}^{m-1} \sigma _j(r_j)^2}\right)\mathop {}\!\mathrm {d}r,\) 

where we denote \(D = [0,d_1(x_1,y_1)/2]\times \ldots \times [0,d_{m-1}(x_{m-1},y_{m-1})/2]\) , \(r = (r_1,\ldots ,r_{m-1})\)  and \(\delta _{x_m,y_m}f(x^{\prime }) = (f(x^{\prime },x_m) - f(x^{\prime },y_m))/d_m(x_m,y_m)\) . Thus, by Fubini's theorem and Jensen's inequality, we obtain that
\(\left|\bar{g}_{x^{\prime },y^{\prime }}(A_1) - \bar{g}_{x^{\prime },y^{\prime }}(A_2)\right|\\ \le 18^{m-1}d_m(A_1,A_2)\int _{D} \int _{A_1}\int _{A_2} \Psi ^{-1}\left(\frac{\kappa _{\delta _{x_m,y_m}f}}{\prod _{j = 1}^{m-1} \sigma _j(r_j)^2}\right)\frac{\upsilon _m(dx_m)\upsilon _m(dy_m)}{\upsilon _m(A)\upsilon _m(B)}\mathop {}\!\mathrm {d}r\\ \le 18^{m-1}d_m(A_1,A_2)\int _{D} \Psi ^{-1}\left(\int _{A_1}\int _{A_2} \frac{\kappa _{\delta _{x_m,y_m}f}}{\prod _{j = 1}^{m-1} \sigma _j(r_j)^2}\frac{\upsilon _m(dx_m)\upsilon _m(dy_m)}{\upsilon _m(A_1)\upsilon _m(A_2)}\right)\mathop {}\!\mathrm {d}r.\) 

We remark that
\(\int _{A_1}\int _{A_2} \kappa _{\delta _{x_m,y_m}f}\upsilon _m(dx_m)\upsilon _m(dy_m) \le \int _{E_m}\int _{E_m} \kappa _{\delta _{x_m,y_m}f}\upsilon _m(dx_m)\upsilon _m(dy_m) = \kappa _f,\) 

and, thus, we obtain
\(\left|\bar{g}_{x^{\prime },y^{\prime }}(A_1) - \bar{g}_{x^{\prime },y^{\prime }}(A_2)\right| \le 18^{m-1}d_m(A_1,A_2)\int _{D} \Psi ^{-1}\left(\frac{\kappa _{f}}{\prod _{j = 1}^{m-1} \sigma _j(r_j)^2}\frac{1}{\upsilon _m(A_1)\upsilon _m(A_2)}\right)\mathop {}\!\mathrm {d}r.\) 
We now denote \(\bar{g}_{x^{\prime },y^{\prime }}(x_m,r) = \bar{g}_{x^{\prime },y^{\prime }}(B(x_m,r))\)  for any \(x_m\in E_m\)  and
\(B(x_m,r) = \lbrace  x\in E_m; d_m(x,x_m)\le r\rbrace \) 
for a certain \(r>0\) . We now fix \(x_m,y_m\in E_m\)  and let \(\lambda _k = d_m(x_m,y_m)2^{-k}\)  for any \(k\ge 0\) . Since \(d_m(B(x_m,\lambda _0),B(y_m,\lambda _0)) = 3 \lambda _0\) , \(\sigma _m(\lambda _k)\ge \sigma _m(\tau )\)  for any \(\tau \le \lambda _k\)  and \(\Psi ^{-1}\)  is increasing, it follows from (REF ) that
\(&\left|\bar{g}_{x^{\prime },y^{\prime }}(x_m,\lambda _0) - \bar{g}_{x^{\prime },y^{\prime }}(y_m,\lambda _0)\right| \nonumber \\ &\hspace{20.0pt}\le 3\times 18^{m-1}\lambda _0\int _{D} \Psi ^{-1}\left(\frac{\kappa _{f}}{\prod _{j = 1}^{m-1} \sigma _j(r_j)^2}\frac{1}{\sigma _m(\lambda _0)^2}\right)\nonumber \\ &\hspace{20.0pt}\le 6\times 18^{m-1}\int _0^{d_m(x_m,y_m)/2}\int _{D} \Psi ^{-1}\left(\frac{\kappa _{f}}{\prod _{j = 1}^{m-1} \sigma _j(r_j)^2}\frac{1}{\sigma _m(\tau )^2}\right)\mathop {}\!\mathrm {d}r \mathop {}\!\mathrm {d}\tau .\) 

Furthermore, by using (REF ) and since \(d_m(B(x_m,\lambda _\ell ),B(x_m,\lambda _{\ell +1})) = \lambda _{\ell } + \lambda _{\ell +1} = 6(\lambda _{\ell +1} - \lambda _{\ell +2})\)  for any \(\ell \ge 0\) , \(\sigma _m(\lambda _k)\ge \sigma _m(\tau )\)  for any \(\tau \le \lambda _k\)  and \(\Psi ^{-1}\)  is increasing, we obtain that, for any \(k\ge 1\) ,
\(&\left|\bar{g}_{x^{\prime },y^{\prime }}(x_m,\lambda _0) - \bar{g}_{x^{\prime },y^{\prime }}(x_m,\lambda _k)\right| \le \sum _{\ell = 0}^{k-1} \left|\bar{g}_{x^{\prime },y^{\prime }}(x_m,\lambda _{\ell +1}) - \bar{g}_{x^{\prime },y^{\prime }}(x_m,\lambda _{\ell })\right|\\ &\hspace{20.0pt}\le 6\times 18^{m-1}\sum _{\ell = 0}^{k-1}(\lambda _{\ell +1} - \lambda _{\ell +2})\int _{D} \Psi ^{-1}\left(\frac{\kappa _{f}}{\prod _{j = 1}^{m-1} \sigma _j(r_j)^2}\frac{1}{\sigma _m(\lambda _{\ell +1})\sigma _m(\lambda _{\ell +2})}\right)\mathop {}\!\mathrm {d}r\\ &\hspace{20.0pt}\le 6\times 18^{m-1}\sum _{\ell = 0}^{k-1}\int _{\lambda _{\ell +2}}^{\lambda _{\ell +1}}\int _{D} \Psi ^{-1}\left(\frac{\kappa _{f}}{\prod _{j = 1}^{m-1} \sigma _j(r_j)^2}\frac{1}{\sigma _m(\tau )^2)}\right)\mathop {}\!\mathrm {d}r \mathop {}\!\mathrm {d}\tau \\ &\hspace{20.0pt}\le 6\times 18^{m-1}\int _{\lambda _{k+1}}^{\lambda _{1}}\int _{D} \Psi ^{-1}\left(\frac{\kappa _{f}}{\prod _{j = 1}^{m-1} \sigma _j(r_j)^2}\frac{1}{\sigma _m(\tau )^2)}\right)\mathop {}\!\mathrm {d}r\mathop {}\!\mathrm {d}\tau .\) 

Letting \(k\rightarrow +\infty \) , we deduce that
\(&\left|\bar{g}_{x^{\prime },y^{\prime }}(x_m,\lambda _0) - g_{x^{\prime },y^{\prime }}(x_m)\right| \le \nonumber \\ &\hspace{20.0pt}\le 6\times 18^{m-1}\int _{0}^{d_m(x_m,y_m)/2}\int _{D} \Psi ^{-1}\left(\frac{\kappa _{f}}{\prod _{j = 1}^{m-1} \sigma _j(r_j)^2}\frac{1}{\sigma _m(\tau )^2)}\right)\mathop {}\!\mathrm {d}r\mathop {}\!\mathrm {d}\tau ,\) 

and, similarly,
\(&\left|\bar{g}_{x^{\prime },y^{\prime }}(y_m,\lambda _0) - g_{x^{\prime },y^{\prime }}(y_m)\right| \le \nonumber \\ &\hspace{20.0pt}\le 6\times 18^{m-1}\int _{0}^{d_m(x_m,y_m)/2}\int _{D} \Psi ^{-1}\left(\frac{\kappa _{f}}{\prod _{j = 1}^{m-1} \sigma _j(r_j)^2}\frac{1}{\sigma _m(\tau )^2)}\right)\mathop {}\!\mathrm {d}r\mathop {}\!\mathrm {d}\tau .\) 

Thanks to (REF ), (REF ) and (REF ), we obtain the desired result.
Corollary REF  follows directly from Lemma REF  by choosing \(E_1 = [0,1]\)  endowed with \(d_1(t,s) = |t-s|^{\alpha _1+1/p}\)  and \(E_2 = B(0,R)\)  endowed with \(d_2(x,y) = |x-y|^{\alpha _2 +d/p}\) . We also use, for both, Lebesgue's measure and set \(\Psi (t) = |t|^{p}\) . It turns out that \(\sigma _1(r) \simeq r^{1/(\alpha _1+1/p)}\)  and \(\sigma _2(r) \simeq r^{1/(\alpha _2/d+1/p)}\) . With the previous choices at hand, we obtain
\(\int _0^{d_1(s,t)/2}\int _0^{d_2(x,y)/2}\Psi ^{-1}&\left(\frac{\kappa _f}{\sigma _1(r_1)^2\sigma _2(r_2)^2}\right)\mathop {}\!\mathrm {d}r_1\mathop {}\!\mathrm {d}r_2\\ &\simeq \kappa _f^{1/p} \int _0^{d_1(s,t)/2}r_1^{-2/(\alpha _1 p+1)}\mathop {}\!\mathrm {d}r_1\int _0^{d_2(x,y)/2} r_2^{-2/(\alpha _2 p/d+1)} \mathop {}\!\mathrm {d}r_2\\&\simeq \kappa _f^{1/p} |t-s|^{\alpha _1-1/p} |x-y|^{\alpha _2-d/p},\) 

where
\(\kappa _f = \int _{[0,1]^2}\int _{B(0,R)^2} \frac{|\square _{(s,x)}f(t,y)|^p}{|t-s|^{\alpha _1p +1}|x-y|^{\alpha _2 p +d}}dtdsdxdy.\) 
We can finally state a Kolmogorov's continuity theorem.
Theorem 4.4 
Let \(p\ge 1\) , \(\alpha _1>1/p\) , \(\alpha _2>d/p\)  and \(f\in \mathcal {C}([0,1]\times \mathbb {R}^d )\)  a random process. Assume that there exists \(\varepsilon _1,\varepsilon _2\in (0,1)\)  such that
\(\vartheta _f : = \sup _{\begin{array}{c}s,t\in [0,1]\\ x,y\in \mathbb {R}^d\end{array}}\mathbb {E}\left[\frac{|\square _{(s,x)} f(t,y)|^p}{|t-s|^{\alpha _1p+\varepsilon _1}|x-y|^{\alpha _2p+\varepsilon _2}} \right]^{1/p} <+\infty .\) 
Then, for any \(\varepsilon _3>\varepsilon _2\) , there exists a positive random variable \(\Upsilon \in L^p(\Omega )\)  which depends on \(f,d,\alpha _1,\alpha _2,p,\varepsilon _1,\varepsilon _2\)  and \(\varepsilon _3\)  such that, \(\mathbb {P}\) -a.s.,
\(|\square _{(s,x)} f(t,y)| \le \Upsilon |s-t|^{\alpha _1-1/p}|x-y|^{\alpha _2 - d/p}(1+|x|+|y|)^{(d+\varepsilon _3)/p},\) 
for all \(t,s\in [0,1]\)  and \(x,y\in \mathbb {R}^d\) .

Let \(\mathcal {A}= \lbrace  x\in \mathbb {R}^d;\; 1\le |x|\le 2\rbrace \)  be an annulus in \(\mathbb {R}^d\)  such that \(\mathbb {R}^d = B(0,1)\cup \bigcup _{j = 0}^{+\infty } 2^j\mathcal {A}\) .
It follows from Corollary REF  and (REF ) that, for any \(j\ge 0\) ,
\(&\mathbb {E}\left[\sup _{\begin{array}{c}t,s\in [0,1]\\ x,y\in 2^{j}\mathcal {A}\end{array}}\frac{|\square _{(s,x)} f(t,y)|^p}{|s-t|^{\alpha _1p-1}|x-y|^{\alpha _2 p - d}(1+|x|+|y|)^{d+\varepsilon _3}} \right]\\ &\hspace{10.0pt}\le \frac{C_{d,\alpha _1,\alpha _2,p}\vartheta _f }{(1+2^{j+1})^{d+\varepsilon _3}}\int _{[0,1]^{2}\times B(0,1+2^{j+2})^2} \frac{du_1du_2 dz_1dz_2}{|u_1-u_2|^{1-\varepsilon _1}|z_1-z_2|^{d-\varepsilon _2}}\\ &\hspace{10.0pt}\le C_{d,\alpha _1,\alpha _2,p}\vartheta _f\frac{2^{(d+\varepsilon _2)(j+2)}}{(1+2^{j+1})^{d+\varepsilon _3}}\int _{[0,1]^{2}\times B(0,2)^2} \frac{du_1du_2 dzdw}{|u_1-u_2|^{1-\varepsilon _1}|z_1-z_2|^{d-\varepsilon _2}}\\ &\hspace{10.0pt}\lesssim 2^{j(\varepsilon _2-\varepsilon _3)}.\) 

Thus, we deduce that
\(&\mathbb {E}\left[\sup _{\begin{array}{c}t,s\in [0,1]\\ x,y\in \mathbb {R}^d\end{array}}\frac{|\square _{(s,x)} f(t,y)|^p}{|s-t|^{\alpha _1p-1}|x-y|^{\alpha _2 p - d}(1+|x|+|y|)^{d+\varepsilon _3}} \right]\\ &\hspace{10.0pt}\le \mathbb {E}\left[\sup _{\begin{array}{c}t,s\in [0,1]\\ x,y\in B(0,1)\end{array}}\frac{|\square _{(s,x)} f(t,y)|^p}{|s-t|^{\alpha _1p-1}|x-y|^{\alpha _2 p - d}} \right]\\ &\hspace{20.0pt}+ \sum _{j = 0}^{+\infty }\mathbb {E}\left[\sup _{\begin{array}{c}t,s\in [0,1]\\ x,y\in 2^{j}\mathcal {A}\end{array}}\frac{|\square _{(s,x)} f(t,y)|^p}{|s-t|^{\alpha _1p-1}|x-y|^{\alpha _2 p - d}(1+|x|+|y|)^{d+\varepsilon _3}} \right]\\ &\hspace{10.0pt}\lesssim 1 + \sum _{j = 0}^{+\infty }2^{j(\varepsilon _2-\varepsilon _3)}<+\infty ,\) 

which ends the proof.
We also recall another standard Kolmogorov theorem.
Theorem 4.5 
Let \(f\in C([0,1];L^{\infty }_{\mathrm {loc}}(\mathbb {R}^d))\)  be a random process and \(p\ge 1\)  such that
\(\sup _{\begin{array}{c}t,s\in [0,1]\\ x\in \mathbb {R}^d\end{array}}\mathbb {E}\left[ \frac{|f(t,x) - f(s,x)|^p}{|t-s|^{\alpha p + \varepsilon }}\right]<+\infty ,\) 
for a certain \(\alpha >1/p\)  and \(\varepsilon \in (0,1)\) . Then, there exists a positive random variable \(\Upsilon \in L^p(\Omega )\)  which depends on \(f,d,\alpha \)  and \(p\)  such that
\(|f(t,x)-f(s,x)| \le \Upsilon |s-t|^{\alpha - 1/p}(1+|x|)^{(1+\iota )/p},\) 
where \(\iota >0\) .

